
Boo1q_Sl£l_____ 



Gopyrig]i0 o _L3il 



COPYRIGHT' DEPOSHY 



THE 



FIELD ENGINEER: 

A 

f&atfog Book of practice 



IN THE 



SURVEY, LOCATION, AND TRACK-WORK OF 
RAILROADS ; 

CONTAINING 

A LAEGE COLLECTION OF RULES AND TABLES, 

ORIGINAL AXD SELECTED, 

APPLICABLE TO BOTH THE STANDARD AND THE 
NARROW GAUGE, 

AND PREPARED WITH SPECIAL REFERENCE TO 
THE WANTS OF 

THE YOUNG ENGINEER. 

BY 

WILLIAM FINDLAY SHUNK, C.E. 



TWENTY-FIRST EDITION, REVISED. 



NEW YORK 
D. VAN NOSTRAND COMPANY 

25 Park Place 
1918 



■ \ 



Copyright, 1890, 1903, 1918, 
By D. VAN NOSTRAND COMPANY, 




APR -3 I9i8 



©CIA494407 
PRICE, S_&j£ NET _ 

i- Ban HoHtrawJ. ® ammm 



/"K^p 



/ 



THE AUTHOR 

Affectionately Dedicates tbfs JSook 

S TO 

ALBERT J. SCHERZER, C.E., 
g)f5 gomrabe anb pear ^rtenb, 

IN TOKEN OF ESTEEM FOR HIS PROFESSIONAL 
ATTAINMENTS AND RESPECT FOR HIS 
MANLY CHARACTER 



PREFACE. 



The author's principal aim in preparing this volume has 
been, as its title indicates, to serve that large class of young 
engineers who, like himself, have not had the advantage of a 
technical education before going out for their livelihood. 
[ The initial chapters are, therefore, given to a compendious 
exposition of those mathematical truths and methods which 
they must needs become familiar with from the beginning. 
Plane Trigonometry, Logarithms, and propositions relating to 
the circle, are tools of the craft in constant use; ready han- 
dling of them is an indispensable condition of excellence. Be 
not discouraged by obscurities and difficulties at the outset; 
light will gradually break on-the scrutinizing eye, and a way 
always open to manful effort. 

These chapters are followed by instructions as to the adjust- 
ment and use of instruments, and hints concerning field rou- 
tine, which it is thought will be found acceptable to the 
inexperienced learner. The same may be said of the articles 
on staking out work, and those on track problems, with which 
the text of the book closes. They have been written with the 
author's own early ignorance in mind, and with a wish to set 
the subjects forth as plainly as possible, disembarrassed of 
hard words in the description, and of unpractical niceties in 
the operation. 

The chapter on field location is believed to include all the 
problems likely to occur. The author, in compiling it, has 
taken those only which have arisen in his own practice, and 
which, therefore, may arise in the practice of others. H.s 

v 



Vi PREFACE. 

own practice having been unusually large and diversified, 
probably the examples given will prove adequate, directly or 
indirectly, to all contingencies. 

No attempt has been made to swell the bulk of the volume 
with imaginary cases; the object being, not to provide barren 
mathematical exercises, but to teach useful knowledge. 

Problems, also, affecting location in its economical aspects, 
— the balancing of physical and financial conditions, equating 
of alternative lines, and the like, — do not come within the 
scope of the work, and are therefore not treated. 

Considerable pains have been spent on the tables. However 
far the young engineer may eventually outgo his teacher as re- 
gards the text of the bx>k, these are implements of his art 
which never become antiquated, and can never fall into dis 
use. Those herein contained which are original will, it is 
hoped, be esteemed worthy of place with their well-approved 
associates. 

The author invites friendly criticism : he would be pleased 
n receive suggestions, both for the improvement of the book, 
and for the cDrrection of possible errors in it, should another 
edition be called for. 

In dismissing the work from his hands, the precarious 
snatches of time occupied in its preparation, by day and by 
night, during the past two years, which might have been more 
agreeably spent in reading, talking, or musing, recur to the 
writer's mind; and the thought arises, To what end or from 
what motive do people undertake these technical labors? Why 
should Forney and Bourne toil to simplify steam for our ap- 
prehension; Nystrom to compile mechanical, Moles worth and 
Trautwine to epitomize civil engineering; Henck to prepare 
his elegant manual of field mathematics; Box to illustrate 
hydraulics; and Shreve, with lucid pen, to make clear for us 
the strains in truss or arch? The ordinary motives, to en- 
deavor here have no place. There is neither fame nor profit 
in these drudging enterprises. At best the author gives name 



PREFACE. Vii 

to his book; he remains impersonal, — known but indirectly, 
and but to a class. How, then, shall we account for his labors? 
I take it, the Father of mankind has not only made our minds 
to hunger for knowledge as our bodies for food, but has also 
imposed upon u< a kindly law of communion, by virtue whereof 
we cannot do otherwise, without violence to generous nature, 
than share with our fellows whatsoever we have learned that 
seems new and useful. Under this law these beneficial works 
would appear to have had their being, and thus pure are they 
from the stain of selfishness. 

Though the present writer would not arrogate equal fellow- 
ship in the eminent brotherhood named, yet he may justly 
claim like pureness from unworthy motive, and certainly feels 
like comfort at heart to that which they must know, for having 
discharged, in what measure it has been laid upon him, the 

divine obligation. 

WM. F. SHUNK. 

Uahwat, N J. 



PEEFACE TO THIL NINTH EDITION. 



Although the writer has thanked individually all who 
have sent him errata from time to time, he cannot regard this 
enlarged edition of the " Field Engineer" as complete with- 
out a general salutation to them. Thanks, too, for pleasing 
evidences, received from many hands, that the book has been 
helpful to those for whom it was intended. 

He would make special acknowledgment to Mr. T. C. 
Mendenhall, Supt. U. S. Coast and Geodetic Survey, and to 
his assistant, Mr. C. A. Schott, for the new astronomical tables 
herein first printed, which were kindly computed for this 
edition in response to a call for much less. 

He would also express his obligation to Mr. Fred. Brooks, 
C.E., Boston, Mass., whose critical suggestion? for the better- 
ment of the text have in the main been adopted. 

These gentlemen are strangers to their beneficiary, He 



Viii PREFACE. 

knows them only by good report, and by these personal 
courtesies. He had no claim on them but that of Saint Paul's 
"one blood." He can pay back thanks alone, — a residual 
debt being left over to do the like by others as they have done 
by him, to the extent of his limited ability. Thus good turns 
go round, and we civilize by mutual service. 

W. F. & 

Harrisburg, Pa m 
March, 1890, 



ABBREVIATIONS. 



-f- Increased by. 

— Diminished by. 

X Multiplied by. 

-f- Divided by. 

= Equal to. 

\ • Since, or seeing that. 

.*. Hence, or therefore. 

: Indicates the quotient of one divided by the other of the 
quantities it connects, called sometimes the ratio of the quan- 
tities. 

: : Indicates an equality of ratios, and connects equal ratios 
In a proportion. Thus, a lb II c I d indicates that a -f- b = c 
-f- cZ; or it may be read, a is to b as c is to d. 

( ) Brackets indicate that the operations embraced by them 
shall first be performed, and the result treated as a single term 
iu the remaining processes required by a formula. Thus, 
a X h -f- {a -f- b) requires that the product of a and b shall be 
divided by their sum. This expression may also be written 

<d> b a T _ • - . 

— — , or a X — r~i> ov b -\ r- 7. If the brackets be omitted 

a-\-b a-\- b a-\-b 

the expression ax$ ~r- a-\-b would mean \-b. 

A 2 . A small secondary figure annexed thus to an expression 
is called its exponent. It requires the principal to which it is 
attached to be used as many times in continued multiplication 
as there are units in the exponent. Thus, A' 2 = A X A ; A 3 
= A X A X A, which is called the cube, or third power, of A. 

V This is called the square root sign : it signifies that the 
square root of the quantity covered by it is to be taken. 

V If preceded by a small secondary figure, called the index, 
as in the marginal figure, it indicates that the cube root of the 
quantity covered by it shall be taken ; and so on. 

A*. If an exponent be fractional, as in the marginal figure, 
it requires that the square root of the third power of the quan- 
tity covered shall be taken, the numerator indicating the 
power and the denominator the root. 

B. M. Bench-mark : any fixed reference point for the level, 

ix 



ABBREYIA TIONS. 

as outcropping ledge, water-table of building, or other perma- 
nent object. Usually a blunt conical seat for the rod, hewn 
on a buttressed tree-base, having a small nail sometimes driven 
flush in the top of it, and a blaze opposite, on which the eleva- 
tion is marked with kiel. 

T. P. Turning-point : usually marked in the field-book. 

P. I. Point of intersection : as of tangents, which are to b© 
connected by a curve. 

A. D. Apex distance: i.e., the distance from the P. I. to 
the point where a curve merges in the tangent. 

P. C. Point of curve : the stake-mark at the beginning of 
a curve. 

P. T. Point of tangent: the stake-mark at the end of a 
curve. 

P. C. C. Point of compound curvature : the stake-mark 
where a curve merges in another of different curvature, turn- 
ing in the same direction. 

P. K. C. Point of reverse curvature : the stake-mark where 
a curve merges in another turning in the opposite direction. 

B. S. Backsight, in transit work ; or the reading of the rod 
to ascertain the instrument height in levelling. 

F. S. Foresight, in transit work; or the reading of the rod 
to ascertain elevations in levelling. 

H. I. Height of instrument : elevation of the level above 
the datum or zero plane. 

H. W. High water. 

L. W. Low water. 



LOGARITHMS. 
I. — II. 



LOGARITHMS. 



i. 

DEFINITIONS AND PRINCIPLES. 

1. The logarithm of a number is the exponent of the power 
to which it is necessary to raise a fixed number to produce the 
given number ; that is to say, it represents the number of times 
a fixed number must be multiplied by itself in order to produce 
an} r given number. 

The fixed number is called the base of the system. In the 
common system, this base is 10. 

It follows from the above, that the logarithm of any power 
of 10 is equal to the exponent of that power. If, therefore, 
a number is an exact power of 10, its logarithm is a whole 
number. 

If a number is not an exact power of 10, its logarithm will 
not be a whole number, but will be made up of an entire part 
plus a fractional part, which is generally expressed decimally. 
The entire part of the logarithm is called the characteristic ; 
the decimal part is called the mantissa. 

2. The characteristic of the logarithm of a whole number 
is positive, and numerically 1 less than the number of places 
of figures in the given number. 

Thus, if a number lies between 1 and 10, its logarithm lies 
between and 1 ; that is, it is equal to plus a decimal. If a 
number lies between 10 and 100, its logarithm is equal to 1 
plus a decimal ; and so on. 

3. The characteristic of the logarithm of a decimal fraction 
is negative, and numerically 1 greater than the 'number of 0's 
that immediately follow the decimal point. 

The characteristic alone, in this case, i« negative, the man- 

3 



4 MANNER OF USING THE TABLES. 

tissa being always positive. This is indicated by writing the 
negative sign over the characteristic: thus, 2.380211 is equiva- 
lent to — 2 + .380211. (See last example, p. 8.) 

4. The characteristic of the logarithm of a mixed number 
is the same as that of its entire part. Thus the mixed number 
T4.103 lies between 10 and 100; hence its logarithm lies be- 
tween 1 and 2, as does the logarithm of 74. 

5. The logarithm of the product of two numbers is equal to 
the sum of the logarithms of the numbers. 

The logarithm of a quotient is equal to the logarithm of the 
dividend diminished by that of the divisor. 

The logarithm of any power of a number is equal to the loga- 
rithm of the number multiplied by the exponent of the power. 

The logarithm of any root of a number is equal to the loga- 
rithm of the number divided by the index of the root. 

6. The preceding principles enable us to abridge labor in 
arithmetical calculations, by using simple addition and sub- 
traction instead of multiplication and division. 



II. 

MANNER OF USING THE TABLES. 

TO FIND THE LOGARITHM OF ANY NUMBER. 

1. First find the characteristic by rule 2, 3, or 4, give* 
above. 

2. Then, if the number be less than 100, look in column N 
of the table for 10 times or 100 times the amount of it; oppo- 
site this multiple, in column O, will be found the mantissa. 

Thus the logarithm of 6 is 0.778151 ; that of 84 is 1.924279. 

3. If the number lie between 100 and 10000, find the first 
three figures of it in column N; then pass along a horizontal 
line until you come to the column headed with the fourth 
figure of the number. At this place will be found the 
mantissa. 

Thus the logarithm of 7200 is 3.857332; that of 8536 is 
3.931254. 



MANNER OF USING THE TABLES. 5 

4. If the number be greater than 10000, place a decimal point 
after the fourth figure, thus converting the number into a 
mixed number. Find the mantissa of the entire part by the 
method last given. Then take from column D the correspond- 
ing tabular difference, multiply this by the decimal part, and 
add the product to the mantissa just found. The principle 
employed is that the differences of numbers are proportional 
to the differences of their logarithms, when these differences 
are small. 

Thus the logarithm of 672887 is 5.827943; that of 43467 is 
4.638160. 

5. If the number be a decimal, drop the decimal point, thus 
/educing it to a whole number. Find the mantissa of the log- 
arithm of this number, and it will be the mantissa required. 

Thus the logarithm of .0327 is 2.514548; that of 378.024 is 
2.577520. 

TO FIND THE NUMBER CORRESPONDING TO A GIVEN 
LOGARITHM. 

6. The rule is the reverse of those just given. Look in the 
table for the mantissa of the given logarithm. If it cannot be 
found, take out the next less mantissa, and also the corre- 
sponding number, which set aside. Find the difference be- 
tween the mantissa taken out and that of the given logarithm; 
annex as many 0's as may be necessary, and divide this result 
by the corresponding number in column D. Annex the quo- 
tient to the number set aside, and then point off from the left 
hand a number of places of figures equal to the characteristic 
plus 1 ; the result will be the number required. If the char- 
acteristic is negative, the result will be a pure decimal, and 
the number of 0's which immediately follow the decimal 
point will be one less than the number of units in the charac- 
teristic. 

Thus the number corresponding to the logarithm 5.233568 
is 171225.296; that corresponding to the logarithm 2.233568 is 
.0171225. 

MULTIPLICATION BY MEANS OF LOGARITHMS. 

7. Find the logarithms of the factors, and take their sum; 
then find the number corresponding to the resulting logarithm, 
and it will be the product required. 



8 MANNER OF USING THE TABLES. 

Example. 
Find the continued product of 3.902, 597.16, and 0.0314728. 

Operation. 
Log. 3.902. . . 0.591287 
Log. 597.16 . . . 2.776091 
Log. 0.0314728 . 2.497930 

1.865314 = log. 73.3354, the product. 

Here the 2 cancels the -\- 2, and the 1 carried from the deci 
mai part is set down. 

DIVISION BY MEANS OF LOGARITHMS. 

8. Find the logarithms of the dividend and the divisor, and 
subtract the latter from the former; then find the number 
corresponding to the resulting logarithm, and it will be the 
quotient required. 

Example 1. 
Divide 24163 by 4567. 







Operation. 


Log. 


24163 . . 


. 4.383151 


Log. 


4567 . . 


. 3.659631 




0.723520 = 1 






Example 2. 


Divide 0.7438 by 12.9476. 






Operation. 


Log. 


0.7438 . . 


. 1.871456 


Log. 


12.9476 . . 


. 1.112189 



2.759267 = log. 0.057447, the quotient, 

Here 1 taken from - gives 2 for a result. The subtraction, as 
in this case, is always to be performed in the algebraic way. 

9. The operation of division, particularly when combined 
with that of multiplication, can often be simplified by using 
the principle of the arithmetical complement. 

The arithmetical complement of a logarithm (written a. c.) 



MANNER OF USING THE TABLES. 7 

is the result obtained by subtracting it from 10: it may be 
written out by commencing at the left hand, and subtracting 
each figure from 9 until the last significant figure is reached, 
which must be taken from 10. Thus 8.130456 is the arithmet- 
ical complement of 1.869544. 

To divide one number by another by means of the arith- 
metical complement, find the logarithm of the dividend and 
the arithmetical complement of the logarithm of the divisor; 
add them together, and diminish the sum by 10; the number 
corresponding to the resulting logarithm will be the quotient 
required. 

Example. 

Multiply 358884 by 5672, and divide the product by 89721. 

Operation. 

Log. 358884 . . . 5.554954 

Log. 5672 . . . 3.753736 

(a.c.)Lo£?. 89721 . . . 5.047106 



4.355796 = log. 22688, the result. 
The operation of subtracting 10 is performed mentally. 

TO RAISE A NUMBER TO ANY POWER BY MEANS OF LOGA- 
RITHMS. 

10. Find the logarithm of tTie number, and multiply it by 
the exponent of the power; then find the number correspond- 
ing to the resulting logarithm, and it will be the power 
required. 

Example. 

Find the 5th power of 9. 

Operation. 

Log. 9 0.954243 

5 



4.771215 = log. 59049, the power. 



TO EXTRACT ROOTS BY MEANS OF LOGARITHMS. 

11. Find the logarithm of the number, and divide it by the 
index of the root; then find the number corresponding to the 
resulting logarithm, and it will be the root required. 



8 MANNER OF USING THE TABLES. 

Example. 
Find the cube root of 4,096. 

Operation. 
Log. 4,096, 3.612360; one-third of this is 1.204120, to which 
the corresponding number is 16, which is the root sought. 

12. When the characteristic is negative, and not divisible by 
the index, add to it the smallest negative number that will 
make it divisible, and then prefix the same number, witl» % 
plus sign, to the mantissa. 

Example. 
Find the 4th root of .00000081. The logarithm of this num- 
ber is 7.908485, which is equal to 8 + 1 908485, and one-fourth 
of this is 2.477121 ; the number corresponding to this logarithm 
is .03: hence .03 is the root required. 

13. Five-figure logarithms are sufficiently accurate for ordi- 
nary railroad field-work. The tables in this book may there- 
fore, as a rule, be used without interpolation. 



PLAICE TEIGO^OMETEl. 
III.— VIII. 



PLANE TRIGONOMETRY* 



in. 

DEFINITIONS. 



1. Plane Trigonometry treats of the solution of plane tri- 
angles. 

In every plane triangle there are six parts, — three sides and 
three angles. When three of these parts are given, one heing 
a side, the remaining parts may be found by computation. 
The operation of finding the unknown parts is called the solu- 
tion of the triangle. 

10 



PLANE TR1GON0METR Y— DEFINITIONS. 



11 




2. A plane angle is measured by the arc of a circle included 
between its sides; the centre of the circle being at the vertex, 
and its radius being 1. The circle, for convenience, is divided 
into 360 equal parts called degrees; 90 of these parts are 
included in a quadrant, which includes one-quarter of the 
circle, and is the measure of a right angle. Each degree is 
further divided into 60 equal parts called minutes, and each 
minute into 60 equal parts called seconds. Degrees, minutes, 
and seconds are de- 
noted by the symbols j_ _b_ 

°, ', /; : thus the ex- 
pression 7° 22' 33" is 
read, 7 degrees, 22 
minutes, and 33 sec- 
onds. 

3. The complement 
of an angle is the dif- 
ference between that 
angle and a right 
angle. 

4. The supplement of an angle is the difference between 
that angle and two right angles. 

5. Instead of employing the arcs themselves, certain func- 
tions of the arcs are usually employed, as explained below. A 
function of a quantity is something which depends upon that 
quantity for its value. 

The sine of an angle is the distance from one extremity of 
the arc enclosing it, to the diameter, through the other extrem- 
ity. Thus P M is the sine of the angle M O A. 

The cosine of an angle is the sine of the complement of the 
angle. Thus N M = O P is the cosine of the angle M O A. 

The tangent of an angle is a right line which touches the 
enclosing arc at one extremity, and is limited by a right line 
drawn from the centre of the circle through the other extrem- 
ity: the sloping line which thus limits the tangent is called the 
secant of the angle. A T is the tangent and O T the secant of 
the angle MO A. 

The versed sine of an angle is that part of the diameter A P 
which is intercepted between the foot of the sine and the ex- 
tremity of the enclosing arc. 

The cotangent of an angle is the tangent of the complement 
of that angle; the co-versed sine and cosecant are similarly 
denned. Thus BT ; , BN, and OT ; are respectively the co- 
tangent, co-versed sine, and cosecant of the angle M O A. 



12 NATURAL SINES, ETC. 

These terms are in practice indicated by obvious contractions; 
as, sin. A for the sine of A, cos. A for the cosine of A, &c. 

6. The above definitions have been made with reference to a 
radius of 1. Any function of an arc whose radius is R is 
equal to the corresponding function of an arc whose radius is 
1, multiplied by the radius R. So also any function of an arc 
whose radius is 1 is equal to the corresponding function of an 
arc whose radius is R, divided by that radius. 

7. Arcs greater than half a circle are not treated in this 
book, for the reasons that they seldom occur in field practice, 
and that any young engineer, conversant with the methods 
herein given, will, it is presumed, have no difficulty in hand- 
ling such exceptional problems. Those curious in trigo- 
nometrical research should consult special treatises on the 
subject. 



IV. 

NATURAL SINES, ETC. 

1. Natural sines, cosines, tangents, or cotangents are those 
which are referred to a radius of 1. They may be used for all 
the purposes of trigonometrical computation ; but it is found 
more convenient, in many cases, to employ a table of logarith- 
mic functions. 



V. 

LOGARITHMIC SINES, ETC. 

1. Logarithmic functions are the logarithms of the natural 
functions treated in foregoing Art. IV., the characteristics 
throughout being transformed by the algebraic addition of 10, 
which has the effect of referring the tabular functions to a 
radius of 10, 000,000, 000 instead of unity, and of thus, it is sup- 



LOGARITHMIC SINES, ETC. 13 

posed, simplifying computation. For the intelligent use of 
these tables, however, it should be remembered that the char- 
acteristic 9 indicates a negative characteristic — 1, being one 
less than 10; a characteristic 7 indicates a negative character- 
istic — 3, and so on. Hence, in figuring with these logarithms, 
as many tens must be subtracted from the final result as have 
been added in the operation. This rule is illustrated in the 
following examples. 

TO FIND THE LOGARITHMIC FUNCTIONS OF AN ARC WHICH 
IS EXPRESSED IN DEGREES AND MINUTES. 

2. If the arc is less than 45°, look for the degrees at the top 
of the page, and for the minutes in the left-hand column; 
then follow the corresponding horizontal line till you come to 
the column designated at the top by sine, cosine, tang., or 
cotang., as the case may be; the number there found is the 
logarithm sought. 

Thus, log. sin. 19° 55' ... . 9.532312 
log. tang. 19° 55' . . . . 9.559097 

3. If the angle is greater than 45°, look for the degrees at 
the bottom of the page, and for the minutes in the right-hand 
column ; then follow the corresponding line towards the left, 
til J you come to the column designated at the bottom by sine, 
cosine, tang, or cotang, as the case may be ; the number there 
found is the logarithm sought. 

Thus, log. cos. 52° 18' . . • . 9.786416 
log. tan. 52° 18' ... . 10.111884 

4. If the arc is expressed in degrees, minutes, and seconds, 
proceed as before with the degrees and minutes ; then multiply 
the corresponding number taken from column D by the num- 
ber of seconds, and add the product to the preceding result, 
for the sine or tangent, and subtract it therefrom for the cosine 
or cotangent. 

Example. 
Find the logarithmic sine of 40° 26' 28". 



14 LOGARITHMIC SINES, ETC. 

Operation. 

Log. sine 40° 26' 9.811952 

Tabular diff. 2.47 
No. of seconds, 28 

Product . 69.16 to be added . 69 



Log. sine 40° 26' 28" 9.812021 






5. If the arc is greater than 90°, find the required function 
of its supplement. Thus the logarithmic tangent of 118° 18' 
25", is equivalent to that of its supplement, or 61° 41' 35". 
and is 10.268732. Also the logarithmic cosine of 95° 18' 24 ;) 
is 8.966078, and the log. cot. of 125° 23' 50" is 9.851619. 

TO FIND THE ARC CORRESPONDING TO ANY LOGARITHMIC 
FUNCTION. 

6. This is done by a reverse process. Look in the proper 
column of the table for the given logarithm; if it is found 
there, the degrees are to be taken from the top or bottom, and 
the minutes from the left or right hand column, as the case 
may be. If the given logarithm is not found in the table, find 
the next less logarithm, take from the table the corresponding 
degrees and minutes, and set them aside. Subtract the loga- 
rithm found in the table from the given logarithm, and divide 
the remainder by the corresponding tabular difference. The 
quotient will be seconds, which must be added to the degrees 
and minutes set aside, in the case of a sine or tangent, and 
subtracted in the case of a cosine or cotangent. 

Example. 
Find the arc corresponding to log. sin. 9.422248. 

Operation. 
Given logarithm . . . 9.422248 
Next less in table . . . 9.421857 . . . .15° 19' 



Tabular diff. . . 7.68) 391 (51" to be added. 

Hence the required arc is 15° 19' 51". 

7. By analogous process, the arc corresponding to log. cos, 
9.427485 will be found to be 74° 28' 43". 



GENERAL PROPOSITIONS. 15 

VI. 
GENERAL PROPOSITIONS. 

1. In any right-angled triangle the hypothenuse is to one of 
the legs as the radius to the sine of the angle opposite to that 
leg. 

And one of the legs is to the other as the radius to the tan- 
gent of the angle opposite to the latter. 

2. In any plane triangle, as one of the sides is to another, so 
is the sine of the angle opposite to the former to the sine of 
the angle opposite to the latter. 

3. In any plane triangle, as the sum of the sides about the 
vertical angle is to their difference, so is the tangent of half 
the sum of the angles at the base to the tangent of half their 
difference. 

4. In any plane triangle, as the cosine of half the difference 
of the angles at the base is to the cosine of half their sum, so 
is the sum of the sides about the vertical angle to the third 
side, or base. 

Also, as the sine of half the difference of the angles at the 
base is to the sine of half their sum, so is the difference of the 
sides about the vertical angle to the third side, or base. 

5. In any plane triangle, as the base is to the sum of the 
other two sides, so is the difference of those sides to the 
difference of the segments of the base made by a perpendicular 
l et fall from the vertical angle. 

6. In any plane triangle, as twice the rectangle under any 
two sides is to the difference of the sum of the squares of 
those two sides and the square of the base, so is the radius to 
the cosine of the angle contained by the two sides. 



16 



SOLUTION OF PLANE TRIANGLES. 



VII. 



SOLUTION OF PLANE TRIANGLES. 

1. It is usually, though uot always, best to work the propor- 
tions in trigonometry by means of logarithms, taking the 
logarithm of the first term from the sum of the logarithms of 
the second and third terms, to obtain the logarithm of the 
fourth term; or adding the arithmetical complement of the 
logarithm of the first term to the logarithms of the other two, 
to obtain that of the fourth. 

2. There are three distinct cases in which separate rules are 
required. 



CASE I. 

3. When a side and an. angle are two of the given parts, the 
solution may be effected by proposition 2 of the preceding 
section. 

If a side be required, say, — 

As the sine of the given angle is to its opposite side, 

So is the sine of either of the other angles to its opposite side. 

4. If an angle be required, say, — 

As one of the given sides is to the sine of its opposite angle, 
So is the other given side to the sine of its opposite angle. 
The third angle becomes known by taking the sum of tin* 
two former from 180°. 

Example 1. 
Given angle A = 24° 26 r ; angle 
B = 36° 43'; side 6 = 137.6: to find 
side a. 




As sin. B . 
Is to sin. A 
So is b . . 



To a, 95.2 



log. 
log. 
log. 



9.776598 
9.616616 
2.138618 



11.755234, sum of 2d and 3d terms, 
log. 1.978636 less 1st term. 



SOLUTION OF PLANE TRIANGLES. 17 

Example 2. 
Given, sides a and 6, as above, and angle A ; to find angle B. 

As side a (a. c.) log. 8.021364 

Is to sin. A log. 9.616616 

So is side b . log. 2.138618 



To sin. B = 36°43 / log. 9.776598 sum. 



CASE II. 

5. When two sides and the included angle are given, the 
solution may be effected by means of propositions 3 and 4. 
Thus, take the given angle from 180° ; the remainder will be 
the sum of the other two angles. 

Then, by proposition 3, — 

As the sum of the given sides is to their difference, 

So is the tangent of half the sum of the remaining angles 
to the tangent of half their difference. 

Half the sum of the remaining angles added to half their 
difference will give the larger of them, and half their sum 
diminished by half their difference will give the lesser of them. 

The solution may be completed either by proposition 4, or 
by proposition 2, as in Case I. 

Examine. 
Given side a =95.2, side 6 = 137.6, and the included angle 
c = 118° 51'; to find the remaining angles. Here 180.00 — 
118° 51' = 61° 09', the sum of the remaining angles. 

As sum of given sides, 232.8 log. 2.366983 

Is to their difference, 42.4 log. 1.627366 

So is tang. $ sum of rem. angles, 30° 34}' . log. 9.771447 

To tang. £ their difference = 6° 08|' . . . log. 9.031830 

Adding half the difference to half the sum, 30° 34|' + 6° 
08^ = 36° 43', = the larger angle, B. Deducting half the 
difference from half the sum =24° 26 r = the smaller angle, A. 

This case is susceptible of solution also by means of propo- 
sition 6. 



18 RIGHT-ANGLED PLANE TRIANGLES. 



CASE TIT. 

6. When the three sides of a plane triangle are given, to 
find the angles. 

First Method. 

Assume the longest of the three sides as base; then say, 
conformably with proposition 5, — 

As the base is to the sum of the two other sides, 

So is the difference of those sides to the difference of the 
segments of the base. 

Half the base added to half the said difference gives the 
greater segment, and diminished by it gives the less; thus, by 
means of the perpendicular from the vertical angle, the original 
triangle is divided into two, each of which falls under the first 
case. Or they may be solved by the simpler methods applica- 
ble to right-angled triangles. 

Second Method. 

7. Find any one of the angles by means of proposition 6, a*id 
the remaining angles either by a repetition of the same rule, 
or by the relation of sides to the sines of their opposite angles. 



VIII. 

RIGHT-ANGLED PLANE TRIANGLES. 

1. Right angles may be solved by the rules applicable to all 
plane triangles; and it will be found, since a right angle is 
always one of the data, that the rule usually becomes simplified 
in its application. 

2. When two of the sides are given, the third may be found 
by means of the rule that the square of the hypothenuse is 
equal to the sum of the squares of the remaining sides. 

3. Another method for solving right-angled triangles is as 
follows: — 

To find a side. Call any one of the sides radius, and write 
upon it the word " radius," Observe whether the other sides 




RIGHT-ANGLED PLANE TRIANGLES. 19 

become sines, tangents, cosines, or the like, and write upon 
them the proper designations accordingly. Then say, 
As the name of the given side is to the given side, 
So is the name of the required side to the required side. 

4. To find an angle. Assume one side to be radius, and 
mark the remaining sides as before. Then say, 

As the side made radius is to radius, 
So is the other given side to the name 
of that side; 
Which determines the opposite angle. 

5. Applying this method to the right- 
angled triangle A B C, and calling the 
hypothenuse a radius, we shall have, 

c = a sin. C -f- R; hence sin. C = Re -f- a. 
b = a cos. C-i-R; hence cos. C = Rb -f- a. 

Then, assuming the side b to be radius, we shall have, 

c = b tang. C -r- R; hence tang. C = Re -f- b. 

If radius be called 1, the natural sines and cosines will be 
used in the application of these formulas; they are often more 
convenient than logarithms in railroad practice, especially 
when the numbers which measure the sides of the triangle are 
either less than 12, or are resolvable into factors less than 12. 

6. The simpler relations between these natural functions are 
as follows : 

. . . 9 . . sin. cos. 1 

sin. 2 -\- cos. 2 = 1; tang. = ; cotanq. = -r— = ; 

cos. sin, tang. 

1 1 . ' . 

sec. = — ; co sec. — —r- ; versme — l — cos.\ co-versme=l — sin. 
cos. sin. 



ADJUSTMENT AND USE 

OF 

INSTRUMENTS. 
IX. -XV. 



ADJUSTMENT AND USE 

OF 

INSTRUMENTS. 



IX. 

GENERAL REMARKS ON ADJUSTMENTS. 

1. Care should be taken in all instrumental adjustments, 
where screws work in pairs, to loosen one before tightening its. 
^ppcsite. 

1. Remember that the eye -piece inverts the image of the 
cross-hairs, and that consequently any movement of it, by 
n>f ans of the small capstan head screws on the outside of the 
telescope-barrel, should be in the direction which would seem 
to increase the error requiring correction. 

3. Before beginning the adjustments, screw the object-glass 
close home, and make a pin-scratch across its rim and the end 
of the tube, by which to mark its proper place; draw out the 
eye-piece until the cross-hairs are exactly in focus; that is to 
say, until no movement of the eye shall appear to displace 
them, and bring the object to be observed clearly into view. 

4. Never permit the glasses to be rubbed with a gritty fabric. 
To remove the dust from them, use a soft, clean handkerchief, 
and change often the part applied. 

23 



24 . THE LEVEL. 

X. 
THE LEVEL. 

tO BRING THE INTERSECTION OF THE CROSS-HAIRS INTO 
THE OPTICAL AXIS OF THE TELESCOPE. 

1. Set the instrument firmly, cast loose the wyes, and, by 
levelling and tangent screws, bring either of the cross-hairs to 
coincide with a well-defined object, distant from 400 to 000 
feet, or as much farther as distinct vision can be had free from 
heat ripple. Gently rotate the telescope half-way around in 
the wyes. If the cross-hair selected for treatment then fails 
to coincide with the object, reduce the error one-half by means 
of the small capstan head screws at right angles to it on the 
telescope-barrel. Bring the spider-line again to coincide with 
the object by means of the levelling and tangent screws, and, 
if necessary, repeat the operation. Proceed in the same man- 
ner with the other cross-hair. If the error is large, bring both 
nearly right before undertaking their final adjustment. 

2. Having thus adjusted the line of collimation upon a dis- 
tant point, requiring the object-tube to be drawn well in, select 
a point close by, which shall require it to be thrust out almost 
to its limit. If any error appears, correct half of it with the 
small screws provided for the purpose, a little forward of the 
diaphragm, and usually protected by a movable sleeve on 
the outside; correct the other half with the levelling-screws. 
After completing this adjustment, test the former one on a 
distant object, and, if necessary, repeat the operations. 

3. In the transit, the small guide-ring screws used for this 
adjustment are covered by the bulb of the cross-bar in which 
the telescope is fixed, and are therefore inaccessible. The 
adjustment, however, is one not liable to become deranged in 
either instrument, and, in the transit, is of comparatively 
small importance. 

4. The young practitioner should beai in mind that the 
intersection of the cross-hairs may coincide with the optical 
axis of the telescope, and yet be out of centre as regards the 
field of view. Such eccentricity does not affect the working 
accuracy of the instrument, which depends upon the position 



THE LEVEL. 25 

of the object-piece solely. It may be removed by manipulation 
of the small screws securing the inner end of the eye-piece. 

TO BRING THE LEVEL BUBBLE PARALLEL WITH THE TELE- 
SCOPE AXIS. 

5. Clamp the instrument over either pair of levelling screws, 
and bring the bubble to the middle of its tube. Turn the tele- 
scope slightly on its bearings, so that the bubble-case shall 
project a little on one side or the other. If the bubble slips, 
correct half its movement by means of the small lateral capstan 
head screws at one end of the case. Return the telescope to 
its first position, level up again, and repeat the operation until 
the erroneous movement ceases. This adjustment brings the 
telescope and level into the same vertical plane. 

G. Next, the bubble being at the middle of its tube, carefully 
lift the telescope out of the wyes, turn it end for end, and 
replace it. If the bubble settles away from the middle, bring 
it half-way back by means of the capstan-heads, working up 
and down at one end of the case. Again middle it with the 
levelling screws, and repeat the operation until the error is 
corrected. 

TO ADJUST THE WYES ; OR, IN OTHER WORDS, TO BRING THE 
TELESCOPE INTO A POSITION AT RIGHT ANGLES TO THE 
VERTICAL AXIS OF THE INSTRUMENT. 

7. Close the wyes. Unclamp. Set the telescope directly 
over two of the levelling screws, and with them bring the 
bubble to the middle of the tube. Then rotate the telescope 
horizontally, until it stands over the same pair of screws, 
changed end for end. If the bubble errs, correct one-half of 
the deviation with the capstan head nuts at the end of the 
bar, and one-half with the levelling screws. Place the tele- 
scope over the other pair of levelling screws. Repeat the 
operation there; and continue the corrections, over one and 
the other pair of levelling screws alternately, until the bubble 
stands without varying during an entire revolution of the 
instrument upon its vertical axis. 

8. The capstan head nuts on the cross-bar should be moved 
by gradual stress, not by pounding. They are a rude device. 
With so short a leverage as the length of the common adjust- 
ing-pin supplies, it is almost impossible to give them a smooth, 



26 LEVELLING. 

manageable motion. Tliey reproach the instrument-maker's 
art as unchecked hydrophobia and cancer do that of medicine, 
or mercenary villany that of law, and should be supplanted by 
better practice. 

9. Having thus completed the principal adjustments in their 
proper order, bring the telescope and its bubble-case as nearly 
vertical in the wye bearings as hand and eye can make them, 
and by reference to a plumb-line, or the corner of a well-built 
house, see if the vertical hair is out of true. If so, slightly 
loosen two opposite screws of the diaphragm, and correct the 
error by turning it. Try again the adjustment of the line of 
collimation before pinning up the wyes. 



XL 

LEVELLING. 



1. Suppose O the starting-point; 1, 2, 3, «fcc, the stakes of 
survey; and A the initial bench-mark. Wherever convenient 
the elevation of A above mean tide should be ascertained. 
It is to be regretted that this was not done from the outset, 




under statute provisions requiring maps and profiles also to 
be filed at the several State capitals. In that case, not only 
would much after labor and expense by way of duplicate sur- 
veys have been spared, but the older Commonwealths would 
now have in hand materials for the preparation of physio- 
graphical maps, the value of which to science, to the engineer, 
and to the economical geologist, it were hard to measure. 



LEVELLING, 



27 



2. For the purposes of a rail road -survey, however, such 
determination is not needful. Any elevation may be assumed 
for A, taking care only that it be large enough to avoid the 
possibility of having minus levels, which would be inconven- 
ient. Zero of the datum should be below the lowest probable 
ground on the contemplated line. 

3. Let the elevation of the initial bench-mark, A, in the 
figure, be taken at +200. Set the level at B, and suppose the 
rod on the BM to read 2.22. The " instrument height" then 
is 202.22. If the rod at sta. O reads 8.4, the elevation at that 
point is 202.22 — 8.4 = 193.8. The rod reading 1.9 at sta. 1, 
the elevation there is 202.2 — 1.9 = 200.3. If desirable to turn 
at sta. 2, drive a pin nearly to the ground at that stake ; sup- 
pose the rod on it to read 0.81. The elevation then is 202.22 — 
0.S1 = 201.41. Now move the instrument to C, and, sighting 
back to sta. 2, let the rod standing on the pin read 2.64. This 
makes the new " instrument height" at C = 201.41, the height 
of sta. 2, + 2.64 = 204.05, and the elevations at 3, 4, 5, or 
other points observed from C are found by deducting the 
readings at those points from the ascertained instrument height 
at the new point of observation. 

4. It thus appears how simple is the rule of levelling, 
namely: Find the "instrument height" by adding the "back- 
sight" to the elevation of the point upon which the rod stands 
for that purpose : from the "instrument height" thus found 
deduct the "foresights," severally, in order to find the eleva- 
tions of the points at which such foresights are taken. 

5. The foregoing example would appear in the field-book aa 
follows : — 



Sta. 


B. S. 


Inst. 


F. S. 


Eleva. 


Remarks. 


BM 








200.00 


B M on W. Oak. 




2.22 


202.22 






40 ft. N. of Sta. O. 





.. 




8.4 


193.8 




1 


.. 




1.9 


200.3 




2 


2*.64 


204 ! 05 


0.81 


201.41 




3 






3.7 


200.3 




4 






3.2 


200.8 




5 


•• 


•■ 


10.36 


193.69 


1 



0. In levelling where great exactness is necessary, the rod at 
turning-points should be read to thousandths, and the reading 
checked by the leveller. Before taking it down, after clamp- 



2g LEVELLING. 

ing the target fast, it should be swayed slowly to and fro in the 
direction of the instrument to make sure of getting the full 
height. In foul weather the rodman should take care that 
the foot of the rod does not ball up with mud or snow. The 
leveller should have his cross-hairs free from parallax, the tar- 
get in focus, and see his bubble true at the moment of obser- 
vation. He should also set the instrument about half-way 
between turning-points when practicable, balancing largely 
unequal sights by subsequent ones similarly unequal in the 
opposite direction ; and his turning-points, even on favorable 
ground, ought not to be more than 600 or 800 feet asunder. 

7. On ordinary railroad field work such nicety as is implied 
in most of these rules is not required. To read to the nearest 
tenth is sufficient, especially where the progress ot the party 
depends in a good degree on the level ; as, for example, in run- 
ning grade lines on preliminary survey. The location levels 
are usually carried along more carefully; but even then the 
writer's practice has been to turn to hundredths only. 

8. The Philadelphia Rod is the best for our service. The 
sliding halves are unconnected except by brass sleeves or 
clips, which guide them, and are therefore not liable to bind ii/ 
wet weather. They are made by William J. Young's Sons 
who some years ago, at the writer's suggestion, supplied what 
seemed to be their only defect by'adopting rivets for fastening 
the clips instead of wood screws: the screws had a tendency te 
work loose in the field, and cause the parts to chafe or jam. 
These rods are clearly figured, so as to be legible at a distance 
of several hundred feet; the leveller is thus enabled to take 
intermediate elevations rapidly, and, when necessary, to d< 
his work with the aid of an unlettered rodman. 

9. CORRECTION FOR THE EARTH'S CURVATURE AND REFRAC- 
TION. 

The correction for a 100-feet "station" is .000215; for one 
mile, 0.6. It is to be added to the calculated elevation of the 
point observed, or to be deducted from the "rod" before 
calculating the elevation, in the case of a long unbalanced 
sight. It varies as the square of the distance. Calling the 
required correction A, for any given distance D, then A — 
.000215 X D 2 if D is in " stations," and A = 0.6 X D 2 if D 
is in miles. Thus the correction for 10 stations would be 



LEVELLING. 29 

.0215; for 50 stations, 0.5375; for 10 miles, 60 feet, and a spire 
Dr treetop apparently level with the instrument at that dis- 
tance would really be (30 feet above it. Transposing the equa- 
tion we have D = ^ A-f-0.6. In this form it is applicable to 
the determination of distances at sea. The Peak of Teneriffe, 
for example, 16,000 feet high, should be just visible from the 
sea-level at a distance = y/ 16000-^-0.6 = say 163 miles. 

10. TO FIND DIFFERENCES IN ELEVATION BY MEANS OF THE 
BAROMETER. 

Call the required difference D; the barometrical reading 
at the lower stand, L; that at the upper stand, U. 

Then, D = (L — U) -*- (L + U) X 55000. 

Example. 
L== 26.64; U = 20.82. 

Then, L — U = 5.82 .... log. 0.764923 
L + U = 47.46 .... log. 1.676328 

0.1226 Diff. —1.088595 

And 0.1226 X 55000 = 6743, the required difference of elevation 
in feet. 

11. A closer approximation is thought to be attainable by 
using a thermometer in connection with the mercurial barome- 
ter. In that case, having found the difference as above, add 
? 1q of the result for each degree by which the mean tempera- 
ture of the air at the two stands exceeds 55° ; subtract the like 
proportion if the mean temperature be below 55°. When the 
upper thermometer reads highest, for "subtract" say "add," 
and vice versa in the foregoing rule. 

12. The naked formula, however, will usually be sufficient 
for the engineer. He can prescribe gradients by it for surveys, 
#hich shall develop the ground to he occupied, and can decide 
between summits well differenced in height. If not so differ- 
enced, questions of detour, of approaches, and the like, will 
contribute to determine the expediency of making an instru- 
mental examination. 

13. HEIGHTS BY THE THERMOMETER. 

T = the difference, in degrees Fahrenheit, between 212°, the 
temperature of boiling water at the sea level, and that at the 
place of observation, 



30 SETTING SLOPE STAKES. 

H = the height of place of observation above or below th« 
sea in feet. 

H = 513 T + T 2 . 

Example. 
T = 212° — 208° = 4°. 
H = (513 X 4) + 4 2 = 2068 feet. 



XII. 

SETTING SLOPE STAKES. 

1. Like swallowing, this is more easily done than described, 
To no detail of field service does the proverb more fitly apply, 
that " work makes the workman. " 

2. The problem is, to find where a formation slope of given 
inclination, beginning at the side of the road-bed, must needs 
intersect the ground surface. Formation slopes are usually 
stated in parts horizontal to one part vertical. Thus a slope 
of 45° is " 1 to 1." A slope of " 2 to 1 " has a horiaontal reach 
of two feet to each foot vertical. The carriages of a stairway 
with twelve-inch treads and eight-inch risers would have a 
slope of "Htol." 

3. To fix the point where any proposed formation slope must 
meet the surface on level ground, is quite simple; the distance 
from the centre line being obviously made up of half the width 
of road-bed added to the horizontal distance due from the 
slope, to the depth of cut or height of fill. Thus, with 20 feet 
road-bed, 9 feet cut, and slope of 1£ to 1, the distance out 
Would be 10 + 9 + 4$ = 23| feet, as shown in the annexed 
diagram. 



if 



4. On slant or broken ground, the solution is more difficult: 
recourse must then be had to the level, with a rodman, a tape* 
man, and, for good speed, an axeman to assist. 



SETTING SLOPE STAKES. 



31 



Example No. 1. 
5. Let the accompanying figure represent the cross-section at 
any given point of a proposed excavation ; road-bed 20 feet wide, 
cutting at centre stake 12 feet, and formation slopes 1 to 1. 



*~ 



"1 

(f- 14.6 *f- 



—T/KTT 



X 




6. The first step is to set the level, as at A, commanding, let 
us suppose, the lower slope, and to ascertain its height above 
grade at the proposed section. This is usually done by refer- 
ence to the nearest bench, and pegging from stake to stake as 
the work progresses. Unless the ground is very steop, and the 
slope-stakes largely different in elevation, labor will be saved 
and likelihood of error reduced by levelling over the centre 
line beforehand, as a separate job, and marking on centre 
stakes the cuts, fills, and grade points, that is to say, the points 
where excavation passes into embankment. The rods should 
be taken carefully at the stakes, and the latter marked on 
their backs to the nearest tenth, as "grade," " C 12," signify- 
ing cut 12 feet, or " F 6.2," signifying fill 6.2 feet, for ex- 
ample. This being done, each centre stake serves as a bench- 
mark for slope staking at that section, and each cross section 
can be staked out independently. 

7. Instrument height, in the example treated, being by either 
method fixed at 15.5 above grade, the next step is a guess how 
far out from the centre stake the formation slope would proba- 
bly meet the ground surface. The closeness of the guess will 
correspond to the experience and natural skill of the leveller; 
the young engineer should not be discouraged if he misses the 
mark rather widely in his early trials. 



32 SETTING SLOPE STAKES. 

8. It is true, that, on a uniform declivity, lie migLt aid con- 
jeeture by taking a rod distant half the width of road-bed, or 
10 feet, from the centre stake, ascertain thus the slope of the 
ground surface as well as the cutting at that point; and with 
these data, knowing also the formation slope, approximate 
the point sought by solving the terminal triangle of the pro- 
posed section, indicated by dotted lines in the figure. But, in 
practice, he will find it the quicker and better way to approxi- 
mate the point by vividly imagining the underground forma- 
tion lines; or by vividly imagining a level section, the upper 
surface of which shall coincide with his instrument height,, 
15.5 feet above grade. This gives him a point in the air,, 
10 -f- 15.5 = 25.5 feet out from the centre stake, level with the 
instrument, as the limit of the imaginary section; and from; 
that point he can pretty well judge where a line corresponding; 
to the formation slope must meet the ground. 

9. Suppose him, by either method, or even by random guess, 
to think that 10 feet for half the road-bed, and 10 more for 
the slope, looks about right. The formation slope being 1 to 
1, this implies a cutting of 10 feet at the side stake, and a rod, 
therefore, of 15.5 — 10.0 = 5.5 feet. Taking a rod accordingly, 
20 feet out, measured horizontally from the centre stake, he 
finds it to be 11.0 instead of 5.5, indicating that he has gone 
too far down hill. Let him now reason that the rod of 11.0 
corresponds to a cutting of 15.5 — 11.0 = 4.5 feet, and that a 
cutting of 4.5 feet corresponds to a distance out of 10 + 4.5 
= 14.5 feet. Try, then, a rod 14.5 feet out. It proves to be 
0.0, corresponding to a cutting of 15.5 — 9.0 = 6.5, instead of 
4.5 feet, and a distance out of 16.5 instead of 14.5 feet. Try, 
next, 16.5 feet out; the rod there, of 10.0 instead of 9.0, shows 
him again to be in error on the down-hill side of his object; 
but the lessening error shows also that he is approaching it r 
and that a few more like trials will reach it. 

10. Recurring to his first error with the 11.0 feet rod, he 
cannot fail to observe after a little practice, since the ground 
ascends thence toward the centre line, that the side stake 
must fall farther out than the point where his second trial w as 
made; that its true position, in fact, divides the distance be- 
tween those points of observation into two parts which are to 
one another directly as the inclinations of the formation slope 
and the ground surface. By degrees he will grow skilful in 
divining this true position, and, becoming meauwhile quick in 



SETTING SLOPE STAKES. 



33 



observation, will place a slope stake on the second or third 
trial, without conscious effort of mind. 

11. Next, suppose the level at B, 25.5 feet above grade, com- 
manding the upper slope. 

Note the change of ground 11 feet out, and take a rod there, 
recording the observation. The cutting at that point is 
25.5 — 9.5 = 16 feet, corresponding to a distance out for the 
side stake of 10 -f 16 = 26 feet, if the ground were level. A 
trial rod 26 feet out reads 7.8, corresponding to a cutting of 
25.5 — 7.8= 17.7 feet, and a distance out for the side stake 
of 10 -f- 17.7 == 27.7 feet, showing that the point sought is still 
beyond. A repetition of such trials will finally fix it; but, as 
in the case of the lower slope, practice will speedily lessen the 
number and abridge the labor of them. 

12. The foregoing section would be noted in the field book 
as follows: — 



Sta. Dis. 


Left. 


Centre 


Right. 


Area. 


C.Yds 


258 


50 


+ 5.8 
15.8 




+ 12.0 


+ 16.0 
11.0 


+ 18.0 
28.0 







Example No. 2. 
13. In the annexed figure, representing an embankment 14 
feet wide on top, with side slopes of 1| to 1, the first thing to 
attract attention is that the instrument is 1 foot below grade, 



30.0 3| 

« 25.0 sj 



^° 



"7IT 



S^W 



b d<^ 



and that, therefore, 1.0 is to be added to all rods, in order to 
find the height of embankment above the points at which rods 
are taken. 

14. Consider the down-hill side. The engineer, with the 
ground in view, and with the height of embankment at the 



34 



SETTING SLOPE STAKES. 



centre stake to aid him in forming an airy image of the pro- 
posed section, judges that the natural surface and the forma- 
tion slopes will meet 30 feet out. Of this distance, 7 feet are 
due to half the road-bed, and 23 feet to horizontal reach of the 
embankment slope. The slope being 1£ to 1, or f, the hori- 
zontal reach of 23 feet corresponds to a vertical height of 
f of 23 = 15.3 feet; and, since the instrument is 1 foot be- 
low grade, to a rod at the supposed embankment base oi 
153 — 1.0= 14.3 feet. But the rod at that point is only 11 
feet, to which, if 1 foot, the distance of instrument below 
grade, be added, the height of embankment would be 12 feet. 
He may then, as in the case of the upper slope in Example No, 
1, try a rod at the distance out corresponding to the 11 feet 
rod, or 12 feet embankment. This distance would be 7 -f- !2 
-4- 6 = 25 feet, where, on trial, the rod proves to be 10 feei, 
instead of 11 feet, corresponding to an embankment height of 
10 + 1 = 11 feet, and to a distance out of 7 + 11 -f- 5.5 = 23.5 
feet. Approximating thus, by shorter and shorter steps, lie 
finally reaches the point sought. 

15. The process in fixing the upper slope stake is similar to 
that used in fixing the lower one in Example No. 1. The 
several steps are designated by small letters in the figure, and 
a detail of them is not thought necessary. 

16. This section would be noted in the field book as fol 
lows: — 



Sta. 


Dis. 


Left. 


Centre 


Right. 


Area. 


C.Yds. 


140 


62 


— 9.4 
2276 




— 6.3 




— 3.2 
12.7 







Example No. 3. 

17. Here is a case, partly in rock excavation, slope J to 1; 
partly in embankment, slope 1£ to 1 ; road-bed 17 feet wide, 9 
feet of which are on the right of the centre line and 8 feet on 
the left. 

18. For the lower slope suppose the instrument height at A 
to be 0.5 feet above grade; centre cutting 2.5 feet. Find first, 
with a G.5 feet rod, the grade point, to left of centre line, 
which proves to be 2.5 feet out. Note it, and set a stake 
there marked "grade." Note also the change of ground 5ft 



SETTING SLOPE STAKES. 



35 



feet out and 10.0 — 6.5 = ,3.5 feet below grade. Then set the 
lower slope stake as in Example No. 2, observing that in this 




case the instrument is above grade, and that its height above 
grade is to be deducted from the rod at any point in order to 
obtain the height of grade above such point, 

19. Move the instrument to B, say 22.5 feet above grade. 
This elevation, if the cutting on that side be deemed to equal 
it, corresponds to a distance out of 9 feet for road-bed added to 
(22.5 -f- 4) for slope; total, 14.6 feet. The trial rod, however, 
at that distance, instead of reading 0, reads 6* feet, indicating a 
cut 22.5 — 6.0 = 16.5 feet deep, and a distance out correspond- 
ing thereto of 9.0 -f- (16.5 -f- 4) = 13.1 feet. Trying again at 
this distance out, the rod reads 7.6 instead of 6 feet, requiring 
a further movement towards the centre line of (7.6 — 6) -f- 4 
= 0.4 feet. Thus by approximations much more rapid than in 
the case of a flatter formation slope, the point is soon fixed. 

20. The field record of the above is as follows: — 



Sta. 


Dis. 


Left. 

i 


Centre 


Right. 


Area. 


C.Yds. 








0.0 ^ 
















— 6.9 


2.5 ! 
— 3.5 f 






4-15.0 






328 


40 


18.3 


+ 2.5 




12.8 












5.5 J 













36 



VERTICAL CURVES. 



XIII. 

VERTICAL CURVES. 

DIAGRAM GIVING THE ORDINATES OF A PARABOLA AT IN- 
TERVALS OF ^ ? TO THE SPAN, THE MIDDLE ORDINATE 
BEING UNITY. 




1. Suppose gradients descending right and left at an equal 
rate from the summit B, and that it is required to truncate the 
summit with a vertical curve extending 150 feet each way. 

A circular arc consuming so small an angle may be treated 
as a parabola, in which the external secant B F is equal to the 
versed sine Fl). Referring to the above diagram, ordinates 4 
and 8 will be seen to correspond to the ordinates between 




chord AC and the curve in this instance, which ordinates 
therefore will be equal to the middle ordinate FD multiplied 
by 0.89 and 0.55 respectively. Adding these multiples to the 
grade elevation at A, the elevations of the intermediate points 
selected will be ascertained. 



VERTICAL CURVES. 



SI 



Example 1. 
Elevation at A = + 94.0; AB = +1 in 100; B C = — 1 in 
100; AD, DC, each = 150 feet or 1.5 stations of 100 feet each. 

Hence BD = U; and FD ==0.75 feet. 

Ordinate 8 = 0.75 X 0.55 = 0.41. 

Ordinate 4 = 0.75 X 0.89 = 0.67. 
Elevation of grade at 8 — 8 = 94.0 + 0.41 = 94.41. 
Elevation of grade at 4 — 4 = 94.0 + 0.67 = 94.67 f 
Elevation of grade at D = 94.0 + 0.75 = 94.75. 



Example 2. 




Elevation at A = + 94.0. AB = -f 1 in 100; BC = — 0.4 
in 100; AH, level; AD, DH, each = 200 feet, or 2 stations, 
divided into 50 feet spaces, the points of division correspond- 
ing therefore to ordinates 3, 6, and 9 of the preceding diagram. 

C IT = 1 X 2 — 0.4 X 2 = 2.0 — 0.8 = 1.2 feet. 

Ascent from A to C along chord AC = CH-f 8 = 1.2-f 
% = 0.15 per 50 feet. 

BE = BD-|CH = 2-0.6 = 1.4. 
.*. FE = 1.4^2 = 0.7. 
Ordinates at 9 — 9 = 0.7 X 0.44 = 0.31. 
Ordinates at G — 6 = 0.7 X 0.75 = 0.52. 
Ordinates at 3 — 3 = 0.7 X 0.94 == 0.66. 
Mid-ordinate = 0.70. 

The elevations then along the chord A C, ascending at the 
rate of 0.15 per 50 feet, will be : — 

A9 6 3 36 9 C 

94.0 94.15 94.30 94.45 94.60 94.75 94.90 95.05 95.20 



3S 



VERTICAL CURVES. 



to which add the ordinates just found: — 

0.0 0.31 0.52 0.0G 0.70 0.GG 0.52 0.31 0.0 
and the grade elevations on the curve will he: — 

94.0 94.46 94.82 95.11 95.30 95.41 95.42 95.36 95.2 



Example 3. 
Elevation at A = + 94.0; AB=+1 in 100 ; 13 C, A H, level. 
AD, BC, each 200 feet divided into 50-feet spaces, the points 




of division corresponding therefore to ordinates 3, 6, and 9 of 
the ordinate diagram C H = 1 X 2 = 2 feet. 

Ascent from A to C along chord A C = C H -f- 8 == 0.25 per 
50 feet. 

BE = B I) — J C II = 1 foot. 
.\ FE = 1 -4-2 = 0.5. 
Ordinates 9 — 9 = 0.5 X 0.44 = 0.22. 
Ordinates $ — 6 = 0.5 X 0.75 = 0.37. 
Ordinates 3 — 3 = 0.5 X 0.94 = 0.47. 
Mid. ordinate = = 0.50. 

The elevations then along the chord A C, ascending at the 
r:ite of 0.25 per 50 feet, will be: — 

A 9 6 3 3 6 9 C 

94.0 94.25 94.5 94.75 95.0 95.25 95.5 95.75 96.0 

to which add the ordinates just found: — 

0.0 0.22 0.37 0.47 0.5 0.47 0.37 0.22 0.0 

And the grade elevations on the curve will be: — 

94.0 94.47 94.87 95.22 95.5 95.72 95.S7 95.97 96.0 



\ r ER TJ CA L CUR VES. 



Example 4. 
Elevation at A = + 94.0; A B = — 1 in 100; BC, AH, 
level; AD, BC, each 150 feet, divided into 50-feet spaces, the 
points of division corresponding therefore to ordinates 8 and 4 
of the initial diagram C H = 1 X 1.5 = 1.5. 




Descent from A to C along chord AC = CH-ffi = 0.25. 

E B = D B — D E = 1.5 — 0.75 = 0.75 

.*. FE = 0.75^2 = 0.375 
Ordinates 8 — 8 = 0.375 X 55 = 0.21 
Ordinates 4 — 4 = 0.375 X 89 = 33 
Mid ordinate = 0.37 

The elevations then along the chord A C, descending at the 
rate of 0.25 per 50 feet, will be: — 

A 8 4 4 8 C 

94.0 93.75 93.5 93.25 93.0 92.75 92.5 

From which deduct the ordinates just found, 

0.0 0.21 0.33 0.37 0.33 0.21 0.0 

And the grade elevations on the curve will be: — 

94.0 93.54 93.17 92.88 92.67 92.54 92.5 

The figures are drawn much distorted, in order to make the 
illustration clear. 

2 With profile paper at hand, a convenient and quite suf- 
ficient determination of the grade elevations on a vertical 
curve may be made by drawing the gradients to a scale of 2 
feet to an inch vertical, and 50 feet to an inch horizontal. By 
means of the curve protractor (Art. XXV. 1) a suitable arc mar 
then be fitted and struck in, and the elevations read off. 



THE TRANSIT. 



XIV. 

THE TRANSIT. 

l/SSbould the vernier and circle plates be out of parallel, — 
should one or the other be sprung, a defect shown by a slight 
rocking motion when the rims are pinched alternately on op- 
posite sides, — the instrument must be sent to the shop for 
repair. This is a common disease of transits in their old age: 
instrument-makers need to study its cause and cure. 

2. TO ADJUST THE LEVEL TUBES. 

IBring the bubbles to the middle of them by means of the 
'levelling screws. Turn the top of the instrument horizontally 
half way round. If the bubbles then err, reduce the error one^ 
half with the small adjusting screws attached to the tubes, 
and one-half with the levelling screws. Repeat until the ad- 
justment is perfect. 

3. TO ADJUST THE VERTICAL HAIR SO THAT IT SHALL RE- 
VOLVE IN A PLANE, AND MARK BACKSIGHT AND FORE- 
SIGHT POINTS IN THE SAME STRAIGHT LINE. 

Try, first, by reference to the corner of a well-built house, 
or to a plumb-line, whether the hair be truly vertical. If it is 
not, loosen the four small capstan head screws on the outside 
of the barrel slightly, and gently tap the topmost one right or 
left, until the adjustment is effected. 

4. Then, after bringing the four screws to a snug bearing 
again, direct the cross-hair to the edge of some well-defined 
object, as a chain pin, or stake, placed 400 or 600 feet distant. 
Upset the telescope, and place a like mark at about the same 

i distance, and level in the opposite direction. Unclamp. Re- 
solve the instrument horizontally on its spindle half way 

» round, and direct the cross-hair to the point first observed. 
Again upset the telescope. If the cross-hair now strikes aside 
from the second mark, correct one-quarter of the error by 
means of the lateral capstan head screws on the barrel, and 



THE TRANSIT. 41 

one-quarter with the tangent screw. Repeat until the adjust- 
ment is effected. An experienced transitman will generally 
prefer to make this adjustment without aid, points in range 
being readily found. 

5. Having thus brought the cross-hair to revolve In a plane, 
it is next to be seen whether the plane in which it revolves is 
truly vertical. To do so, set the instrument near the base of 
some lofty point, as a church spire or chimney, on which point 
direct the cross-hair, and then, tilting the object end of the 
telescope downwards, set a pin, or make a pencil dot in line. 
Unclamp the spindle; turn the instrument horizontally half- 
way round; clamp fast; fix the cross-hair again on the lower 
point, and try it on the upper one. If it misses, correct half 
the error by means of the adjusting screws now usually pro- 
vided, at one of the hearings of the cross-bar; or, if these be 
lacking, by filing off the feet of the standard which supports 
the higher end of the cross-bar. 

6. TO ADJUST THE NEEDLE. 

Having removed the cap, and placed the instrument con- 
veniently in a still room, push one end of the needle a little 

j aside from the point where it tends to settle, and exactly to 
some figured division line on the graduated circle. There 

i gently stay it in position by means of a small wooden hlock, 
an ivory die, or the like. Observe where the opposite end 

| strikes. If between graduation lines, mark the precise spot 
with a sharp pencil. Turn the needle end for end, and stay 

I the reverse point at the division line first observed. Again 
spot with the pencil where the opposite end stakes. Midway 

j of these two pencil spots make another. Take the needle off 
the pivot, and bend it this way or that, until, hy repeated 

j trials, when replaced with one end stayed at the division line 

J first observed, the other shall cut the midway pencil spot. 

7. The needle being thus straightened, proceed to rectify the 
position of the centre pin, if necessary, by bending it with nip- 
pers so that the needle shall Out opposite degrees at the quarter 
points of the circle. 



MISCELLANEOUS. 



o _ 
O 

2_: 
1 = 

6 = 


— 8 

— 6 

_— 4 

— 2 

- 



XV. 

MISCELLANEOUS. 

THE VERNIEK. 

1. The vernier in the transit is a short 
graduated arc, movable around the graduated 
circle of the instrument, by means of which 
subdivisions of the circle graduation can be 
read. There are many varieties of the ver- 
nier; but a knowledge of the principle upon 
which one is made introduces the student to 
an easy acquaintance with all. 

2. Suppose the tenth part of a foot to be 
marked off on a straight edge into ten equal 
parts, and that on another straight edge a 
space equal in length to nine of. these parts is 
divided also into ten equal parts. The sub- 
divisions of the latter scale will then each be 
nine-tenths as large as the subdivisions of the 
former; and if the graduated edges are placed 
together, with the zero marks in both exact- 
ly lined, the first mark of the latter, or ver- 
nier, scale will fall short of the first mark of 
the former, or limb, so to speak, by one-tenth 
part of the first space on the limb; that is to 
say, by one-tenth part of one-hundredth of 
a foot, or one-thousandth of a foot. The sec- 
ond mark of the vernier will fall short of the 
second mark of the limb by two-thousandths 
of a foot, and so on. If, therefore, the ver- 
nier scale be moved slowly forward, the suc- 
cessive oppositions of the scale marks will 
indicate successive advances of the vernier, 
each equal to the one-thousandth part of a 
foot. The marginal example reads 6.217 = six 
feet, two-tenths, one-hundredth, and seven- 
thousandths. 

3. The annexed figure represents the transit 



MISCELLANEOUS. 



43 




vernier, together with a part of the graduated circle. The 
vernier is a double one, for con- 
venience in reading angles right 
or left. It will be observed that 
a space, equal to twenty-nine half 
degrees on the limb, is laid off 
from zero each way on the ver- 
nier, and there subdivided, on 
both sides of zero, into thirty 
equal parts. If now the zeros 
are brought into line, the first 
marks of the vernier right and 
left will fall one-thirtieth part of 
a half degree short of the first, or 
half-degree marks on the limb; 
that is to say, one minute short. 
The vernier, therefore, is scaled 
to read minutes; and, if its zero 
mark be moved slowly half a 
degree on the limb, its several 
subdivision marks, one after an- 
other in arithmetical succession, 
will be seen to line with marks 
of the limb until the thirtieth is 
reached, when zero will be found 
to have traversed the half degree 
space. 

4. TO READ AN ANGLE. 

First note whether the vernier 
has been moved right or left; 
then observe on the limb the 
number of full degrees, and the 
half-degree, if any, which zero of 
the vernier has passed; next, 
look along the vernier from its 
zero towards the right, if the 
movement has been towards the 
right, and from zero towards the 
left, if the movement has been 
towards the left, until a "minute" mark is found exactly in 
line with some mark on the limb. Add the number of that 



Eli 




44 MISCELLANEOUS. 

minute mark on the vernier to the angle already ascertained 
within half a degree from the limb: the sum will be the angle 
sought. The vernier in the figure reads 1° 20' L. 

5. In some respects a vernier graduated decimally would be 
more convenient on railroad locations, where the 100-feet chain 
is used; the calculation of engineers' tables to sixtieths of a 
degree lias prevented its adoption. 

6. TO RE-MAGNETIZE A NEEDLE. 

Lay the north half flat on a smooth, hard surface, and 
with gentle pressure draw the south pole of a common magnet 
over it, from the centre outwards, withdrawing the magnet 
from it six or eight inches after each pass. Repeat ten or a 
dozen times. Treat the south half of the needle in the same 
manner with the north pole of the magnet. Replace the bal- 
ancing wire. If the needle yet proves to be sluggish, take out 
the centre pin, and newly point and polish it. 

7. If the needle, by reason of electricity, clings to the cover- 
ing glass in the field, a touch of the moist finger to the top of 
the cover will release it. 

8. Do not suffer idlers to play it about with knives, keys, 
and the like. 

9. When the instrument is out of use, leave the needle free. 

10. TO REPLACE CROSS-HAIRS. 

Take out the eye-glass tube. Remove the small lateral 
capstan head screws which hold the cross-hair ring athwart 
the barrel. Loosen the vertical screws, and, taking care 
throughout to observe the position of the ring, in order that it 
may be got back again right side up and right face forward, 
turn it lengthwise of the barrel. Insert the end of a pine 
sliver into one of the side holes, take out the vertical screws, 
and withdraw the ring. Stretch across new hairs, in the scores 
traced for them, of the finest clean spider-line ; secure them 
with a touch of gum or wax, and put the ring in by a reverse 
process. 

11. TO FIX A TRUE MERIDIAN. 

By equal shadows of the sun. 

On level ground or ice, set up a pole. Two or three hours 



MISCELLANEOUS. 45 

before noon, mark the extremity of its shadow. With radius 
reaching to that mark, from a centre on the surface vertically 
below the top of pole, strike an arc eastward. Two or three 
hours after noon, watch for the moment when the extremity 
of the shadow touches the arc. There make another mark. 
The true meridian will pass from the centre midway between 
the two marks, if the observations be made about the period of 
the solstice, in June or December. The method gives a fair 
approximation at any time of year. 

12. By observation of the North Star in meridian. 

Polaris, or the North Star, being not exactly at the pole, re- 
volves around it through a small 
circle. It is therefore due north » ' o 

of an observer only when verti- • • i 

cally above or below the pole. To CASSidpF i A 
observe it at either of these points, • 

reference is had to certain bright • 

stars which are in vertical range \ 

with it near the time of culniina- ■ 

tion. Its vertical range with either Pole ' STAR 

of the reference stars being ob- "7'poLp: "~ ™ 

served, the true meridian may be \ 

set out by means of a direct obser- • 

vat ion of Polaris at an interval ! 

of time thereafter derived from ! 

the accompanying table. ' 

The stars thus used are, first, \ % 
the middle star of the three com- D i P P E ^ * 
posing the handle of the Dipper, ' % • • 
called £ Ursce Mujoris; second, the tg 
star called 8, at the foot of the • * 
first stroke of the W in the con- 
stellation Cassiopeia, which lies I 
opposite the Dipper, at about an I 



equal distance from the pole. Of course, when one of these 
stars is in upper, the other is in lower culmination; and the 
approximate time for observation may be found in Table I.,. 
giving the culminations of Polaris. At present, Jan. 1, 1890,. 
the Pole Star culminates not quite one minute earlier than it 
comes to the same vertical with £ Ursae Majoris— a fact indi- 
cated by the negative sign in the annexed table. The two* 
stars will culminate together two and a half years hence. 



46 MISCELLANEOUS, 

The following table gives, with sufficient accuracy for any 
latitude in the United States south of Alaska, and for either 
the upper or the lower culminations of these bright stars, the 
value of the time interval, and the annual increase thereof in 
minutes, between the moment of vertical coincidence, and 
the moment of the culmination of Polaris. 

Min. Min. 

For ? Ursce Majoris in 1890 - 0.9 ) A , . , A OEr 

in 1900 + 2 6 \ Annual urease -f- 0.35 

For d Cassiopeia in 1890 + 0.1 ) * i . , A on 

in 1900 4- 3 4 \ Annual increase -f- 0.33 

To establish the meridian, c 1 " :ose still weather, hang a 
plumb-bob from some high fixed object into a bucket of 
water, that it may be both free and steadfast, and select a 
place of observation so far southward that the plumb-line 
shall cover the breadth of sky between the reference star and 
the pole, — the farther the better. The point of observation 
may be an upright bodkin or compass-sight, fastened to a 
block movable horizontally eastward and westward. Watch 
for the moment when, from the point of observation, the 
plumb-line covers Polaris and the reference star. On the 
lapse of the tabular interval thereafter bring the plumb-line in 
range w 7 ith Polaris by shifting the observation point laterally 
That range will be the true meridian. Stakes may be set od 
it forthwith by means of candles. 

If the star in Cassiopeia be used within the coming two and 
a half years, attention is directed to the negative time interval. 
Its treatment hardly needs exposition. 

With a transit the plumb-line is not necessary, but special 
care should be taken to adjust the vertical thread of the tele- * 
scope, and the horizoutality of its transverse axis. This is 
best done by sighting up and down a fine cord or wire sus- 
pending a plummet in water. When making observations at 
night the cross hairs maybe illuminated by reflecting light on 
the object glass from white paper. 

13. By observation of the North Star at its extreme elonga- 
tion. 

Find the time in Table II., and make the preparations above 
directed. Keep the plumb-line in range with the star until 
the star apparently ceases to move. Mark that range. Multi- 
ply the natural tangent of the azimuth, given in Table III., by 



MISCELLANEOUS, 47 

the distance in feet from the point of observation to the mark 
in the northern range just set. The product will be the dis- 
tance from said northern range mark, square right or left, to 
a point in the true meridian passing through the point of ob- 
servation. If the western elongation was observed, set off the 
calculated distance eastward from the northern range mark; 
if the eastern elongation was observed, set the distance off 
westward. If both the eastern and western elongations be 
observed, the true meridian will pass through the point of 
observation, bisecting the angle between the northern range 
marks. 

With a vernier instrument, the azimuth can be laid off 
directly, in degrees and minutes. 



PROPOSITIONS AND PROBLEMS 
RELATING TO THE CIRCLE. 



XVI. 

PROPOSITIONS RELATING TO THE CIRCLE. 

The following propositions, demonstrable by simple processes 
of geometrical reasoning, may be regarded as axiomatic. 




1. In any circle a tangent is perpendicular to radius at the 
tangent point. Thus, B I is perpendicular to BO. 

41) 



50 PROPOSITIONS RELATING TO TOE CIRCLE. 

2. Tangents drawn to a circle from the same point are equal 
Thus, I B = I E. 

3. The angle DIE, at the intersection of tangents, is equal 
to the central angle 13 C E, included between radii to the tan- 
gent points. 

4. If a chord BE connect the tangent points, the angles 
I B E, 1 E B, are equal : each of them is equal to half of the 
central angle BC E, or of the intersection angle DIE. 

5. Any angle, BCE, at the centre, subtended by the chord 
BE, is double the angle BFE, at the circumference, on th* 
same side of the chord B E. 

6. Acute angles at the circumference, subtended by equal 
chords, are equal. 

7. An acute angle, KFH, between a tangent and a chord, 
is called a tangential angle, and is equal to the periphera 1 
angle LFH subtended by an equal chord; each is equal to 
half the central angles FCII, or HCL, subdivided by the 
same chords. 

8. The exterior angle LHN at the circumference, between 
two equal chords, is called a deflection angle : it is equal to the 
central angle, or to twice the tangential angle, subtended by 
either chord. 

9. If FKbe made equal to FH, and HX be made equal to 
HL, HK is called the tangential distance, and LN the deflec- 
tion distance. 

10. The exterior angle EHXat the circumference, between 
two unequal chords, is equal to the sum of their tangential 
angles, or to half the sum of their central angles. 



XVII. 

CIRCULAR CURVES ON RAILROADS. 

1. The circle is divided, for convenience, into 360 equal 
parts, called degrees. A circle 36,000 feet in circumference 
would be cut by such subdivision into 360 parts, each 100 feet 
long, and subtending an angle of one degree at the centre; its 
radius would be 5,729.6 feet, usually reckoned 5,730 feet. The 






CIRCULAR CURVES OX RAILROADS. 51 

chain 100 feet long being the unit generally adopted by Ameri- 
can engineers 'for field measurements, any circular arc having 
that radius, of 5,730 feet, is called a one-degree curve, for the 
reason that one chain is equivalent to an arc of one degree at 
the circumference. 

2. The circumferences of circles vary directly as their radii: 
hence, in any circular arc struck with half that radius, or 
2,S65 feet, one hundred feet at the circumference would sub- 
tend an angle of two degrees at the centre. Such an arc is 
called a two-degree curve. If one-third of the primary radius 
of 5,730 feet, or 1,010 feet, be used, the arc is called a three- 
degree curve; and so on. 

3. It should be borne in mind, however, that these measure- 
ments are supposed to be made around the arc itself, and no* 
on lines of chords. Since field measurements with the chain 
are always made on the lines of the chords, which are shorter 
between given points at the circumference than the lines of 
the arcs, as a bowstring is shorter than the bow, it is plain 
that, in advancing towards the centre of the one-degree curve 
by a series of concentric circles having radii equal to one-half, 
one-third, &c, of the primary radius, the chord 100 feet long 
differs more and more in length from the arc subtended by it, 
the bow being more and more arched in relation to the string. 
Thus, in the circle having a radius equal to one-twentieth of 
the primary radius, the chord 100 feet long subtends an angle 
of 20° 00', at the centre, instead of 20°, and the arc is 100.5 
feet in length, instead of 100 feet. In order, therefore, that 
the chord of 100 feet may subtend arcs of 1°, 2°, 3°, &c, in 
regular succession, the radii of these successive arcs must be 
somewhat greater than the above method by subdivision of the 
primary radius would make them; though, as might be inferred 
from the extreme case given by way of illustration, the dif- 
ference is not appreciable in ordinary field practice, and radii, 
together with all the functions dependent on them, may 
usually be held to vary as the degree of curvature, or centra] 
angle per 100 feet chord, varies. 



TO FIND THE RADIUS OF A CURVE. 



XVIII. 

TO FIND THE RADIUS, THE APEX DISTANCE, THE 
LENGTH, THE DEGREE, ETC., OF A CURVE. 

1. Let D B, A L be two straight lines intersecting at D. Lay 
off equal distances, DA, DB; erect perpendiculars at A and 

B, meeting at G, and con- 
'L nect A B, D G. From the 

centre G, with radius G A, 
draw the curve A H B. 

The point D will be the 
P. I.; A and B, tangent 
points; D A, D B, the tan- 
gents, or apex distances, 
which denote by AD; D H, 
the external secant, or S; 
HN, the middle ord, 01 
O. Let the long chord 
AB, connecting the tan- 
gent points, be called C, 

and G A or G B, the radius, R. Call the deflection angle to a 

chord of 100 feet D, as before. 

2. By XVI. 3and4, angle DAB=DBA = AGD = DGE 
= il. 

3. GIVEN THE INTERSECTION ANGLE I AND RADIUS R, TO 
FIND THE APEX DIST. AD. 

A D = R X tan. \ I. 

Example. 
R= 1,910.1, I = 35° 24'. 

Then AD =R tan. i I = 1,910.1 X 0.3191 = 609.5. 

4. Measure from the P. I. equal distances, D M, D F, along 

the tangents. Measure, also, M F and D K, the distance from 

D to the middle point of M F. Then, by reason of similarity 

in the triangles M D K, D A G, 

MK:DK::AG:AD::R:T 
.\AD=RxDK-r-MK. 




TO FIND THE RADIUS OF A CURVE.. 53 



Example. 




MK = 190.5, DK = 60.8, R = 1910.1. 




Then R = 1910.1 .... 


3.281056' 


DK = 60.8 .... 


1.783904 


MK= 190.5 (a.c.) . 


7.720105> 


AD = 609.6 .... 


2.785065 



5. If 100-feet chords be used, find the ap. dist. in Tables 
XVI. corresponding to the given angle I. Divide that tabular' 
ap. dist. by the degree of curvature corresponding to the; 
given radius: the quotient will be the required ap. dist. Thus,, 
Tab. A D corresponding to 35° 24' = 1,828.7, which, divided! 
by 3, the degree of curvature, gives 609.6, the ap. dist. sought,. 

6. GIVEN THE INTERSECTION ANGLE I AND AP. DIST. A D, 
TO FIND RADIUS R. 

Transposing the equation in (3), 

R = A D -f- tan. \1— AD Xcot. $1 

Example. 
AD=609.6, 7=35° 24' B=A D cot. i 7=609.6 X 3.1334=1910.1 

By a like transposition of the equation in (4), 

R= ADxMK+DK 

7. If 100-feet chords be used, find in Table XVI. the ap. 
dist. corresponding to the given angle I. Divide that tabular 
datum by the given ap. dist. ; the quotient will be the degree 
of curvature in degrees and decimals. The radius corre- 
sponding to this degree of curvature may be found by (12), by 
Table X., or, with sufficient accuracy for ordinary practice, by 
dividing 5,730, the radius of a 1° curve, by it. 

Thus, in the foregoing example, the tabular ap. dist. cor- 
responding to 35° 24' is 1,828.7. Dividing by 609.6, we have3 
for the degree of curvature; and 5,730 divided by 3 gives 
R= 1,910 feet. 






54 TO FIND THE RADIOS OF A CURVE. 



%. GIVEN THE INTERSECTION ANGLE I AND CHORD A B = C 
CONNECTING THE TANGENT POINTS, TO FIND RADIUS R. 

AJSr = iAB = iC;AG.]Sr = lL 
A G = A JST -4- sin. A G N ; 

i.e., B = i C -r- sin. -J L 

Example, 
1=35° 24', (7=1161.4. 

Then B = \ C -f- sin. \ J, = 580.7 -f- 0.304 = 1910.2. 

9. If 100-feet chords be used, find in Table XYI. the chord 
corresponding to the given angle I. Divide that chord by the 
given chord, for the degree of curvature in degrees and deci- 
mals. Determine the corresponding radius by (17), by Table 
X., or, for ordinary practice, by dividing 5,730 by it. 

Thus, in the foregoing example, the tabular chord corre- 
sponding to angle 35° 24 ; would be 3,4S4.2, which, divided by 
the given chord, 1,161.4, gives 3 for the degree of curvature, 
and 5,730 divided by 3 makes the radius R = 1,910 feet. 



10. GIVEN THE INTERSECTION ANGLE I AND THE DEGREE 
OF CURVATURE OR DEFLECTION ANGLE D, WITH 100-FEET 
CHORDS, TO DETERMINE THE LENGTH OF THE LONG CHORD 
C, THE MIDDLE ORD. O, THE EXTERNAL SECANT S, OR 
THE APEX DIST. A D. 

Take from the proper column in Table XYI., the number 
corresponding to the intersection angle, and divide it by the 
degree of curvature: the quotient will be the length required. 

Example. 

A 4° curve, 1 = 50° 16'; to find the several functions above 
named. 

Table XYI. gives the designated functions of a 1° curve as 
follows: C 4,867.3, O 542.4, S 599.3, AD 2,688.2. Dividing 
by 4 the degree of curvature, we have for the corresponding 
functionsof a 4° curve as follows: C 1,216.8, O 135.6, S 149.8, 
AD 672.0. 



RADII, DEFLECTION ANGLES, ETC. 55 



11. GIVEN C, O, S, OR A D, OF ANY CURVE, AND D, THE DE- 
GREE OF CURVATURE, TO FIND THE INTERSECTION ANGLE, I. 

Multiply the given function C, O, S, or AD, by the degree 
of curvature, D: the product will be found in the proper col- 
umn of Table XVI., corresponding to the required angle. 

Example 1. 
Given AD = 515, D = 5°; to find I. 

Then A D X D = 2,575, which corresponds in Table XVI. 

to 48° 24' = I. 

Example 2. 

Given C = 1,656, D =n 3°; to find I. 

Then C X D = 4,968, which corresponds in Table XVI. to 
51°23'=L 

12. GIVEN C, O, S, OR A D, OF ANY CURVE, AND THE INTER- 
SECTION ANGLE I, TO FIND THE DEGREE OF CURVATURE D. 

Take from the proper column of Table XVI. the number 
corresponding to the given angle I, and divide that tabular 
number by the length of the given part; the quotient will be 
D, in degrees and decimals. 

Exampte 1. 
Given A D = 587, I = 22° 26'; to find D. 
The A D corresponding to I in Table XVI. is 1,136 JL^, 

Then 1,136.3 -f- 587 = 1.935 = 1° 56' = D. 

Example 2. 
Given S = 64, 1 = 30° 25', to find D. 
The Ex. Sec. corresponding to I in Table XVI. is 208. 

Then 208 H- 64 = 3.25 = 3° 15' = D. 

13. GIVEN THE INTERSECTION ANGLE I, AND DEFLECTION 

ANGLE D, TO FIND THE LENGTH OF THE CURVE. 

Divide I by D: the quotient will be the number of chord 
lengths in the curve. 

If the degree of curvature is a whole number, the more con- 
venient method of effecting the division is, first, to reduce the, 



56 



RADII, DEFLECTION ANGLES, ETC. 



minutes, if any, in I to decimals of a degree; then divide by 
the degree of curvature. 

Example 1. 
I = 20° 40', D = 3°. 20° 40' is equivalent to 20.67 degrees. 
Dividing by 3, we have 6.89 chord lengths for the length of the 
curve. If the chords, as is usual, are each 100 feet long, the 
length of the curve in this case will be 689 feet. If the chord 
lengths were 50 feet each, the length of the curve would be 
half this number of feet. 

14. If the degree of curvature is fractional, the more con- 
venient method of effecting the division is, first, to reduce 
both I and D to minutes; then divide the former by the latter. 

Example 2. 
I = 30° 22', D = 2° 45'. These are equivalent, respectively, 
to 1,822 and 165 minutes. Dividing the former by the latter, 
we have 1,104 feet for the length of the curve. 

15. The ingenious assistant who will attentively consider 
the preceding figures cannot fail to detect other obvious analo 
gies which it has not been thought necessary to include in this 
compendium. 

16. In railroad field practice it is usually sufficient to deter- 
mine angles to the nearest minute, and distances to the nearest 
foot. The nicety of seconds and tenths appears generally to 
be quite superfluous; the time consumed on them were better 

employed in pushing ahead. 



17. GIVEN ANY DEFLECTION AN- 
GLE D, AND CHORD C, TO FIND 
RADIUS R. 

FB -f- sin. iALB = BL; i.e., 
iC-S-sin. £ D = B. 

Example. 
Let C=100 feet, Z) = 4°. 







Then R = $ C -±- sin. J D = 
.0349 = 1432.7. 



:50-r 



If the chords are 100 feet long, as is usual in railroad prac- 
tice, radius may be found with sufficient accuracy by dividing 



g 



RADII, DEFLECTION ANGLES, ETC. 57 

5,730, the radius of a 1° curve, by the deflection angle, or de- 
gree of curvature. Tims, in the foregoing example, 5,730 -5- 4 
= 1,432.5. 

IS. GIVEN ANY RADIUS K, AND CHORD C, TO FIND THE DE- 
FLECTION ANGLE D. 

From the preceding equation and example: — 

Sin }D = iC-7-R = 50-r- 1,432.7 = .0349 = sin 2° = | D 
... D = 4°. 



19. GIVEN ANY RADIUS R, AND CHORD C, TO FIND THE DE- 
FLECTION DISTANCE d. 

First find the deflection angle by above method (18). Then, 
ingle II A B in the figure being made equal to D, and HA 
«= B A, BH will be the deflection distance. Draw AK to the 
aiiddle point of II B. 

Then cZ = HB = 2 KB = 2 AB X sin KAB = 2 C X sin 

Example. 

Let R = 1,146 feet, C = 100 feet. 
By (18) D will be found = 5°. 

Then d = 2C X sin \ D = 200 X .0436 = 8.72 feet. 

20. If the chords are 100 feet long, as is usual in field meas- 
urement, divide the constant number 10,000 by the radius in 
feet: the quotient will be the deflection distance. The deflec- 
tion distance with radius of 10,000 feet and chord of 100 feet 
is one foot: this rule is based upon the principle that deflection 
distances, the chord length being fixed, will vary inversely as 
the radii. 

Thus, in the foregoing example, 10,000 -i- 1,146 = 8.72. 



21. GIVEN ANY RADIUS R, AND CHORD C, TO FIND THE TAN- 
GENTIAL ANGLE T. 

The angle T is equal to \ D by construction; for mode of 
determining it, see preceding section (18). 



58 



ORDINATE 'S. 



22. GIVEN ANY RADIUS R, AND CHORD C, TO FIND THE TA3- 
GENTIAL DISTANCE /. 

First find the tangential angle, as above directed. Then, 
angle 13 A E in the figure being made equal to T, and AE = 
AB, BE will be the tangential distance. Draw AX to the 
middle point of BE. 

Then « =EB = 2XB = 2AB X sin NAB = 2 C X tin 
*T. 

Example, 

Let R = 1,140 feet, C = 100 feet. 
By sect. 1, T will be found = 2° 30'. 

Then i = 2CX sin } T = 200 X .0218 = 4.3G feet. 

23. In ordinary railroad practice the tangential distance Jnay 
le considered equal to half the deflection distance. 



XIX. 

ORDIXATES. 



1. GIVEN ANY RADIUS R, AND CHORD C, TO FIND THE MID 
DLE ORDINATE M. 



r 







N / 


E 

K \ 




G 




L 







In the annexed figure, II X = M, H G = R, A B = C. 

N G=\/A G 2 — A N* = \/K a — i C 2 ; II X = II G- N G. 

i.e., M = R — VR 2 — iC 2 . 



ORD [NATES. 59 

Example. 

R = 819, C = 10l); to find the middle ordinate, M. 

M = 819 — V070761--2500 = 1.53. 

2. Ansle IIAN = i IIGB; IIGB = | AGB, .\ HAN = 

JAGB. 

II X = AN X Ian. II AN; i.e., M=]CX tan. J D; D 
being the central angle subtended by the chord. 

Example. 
D = 7°, C = 100; to find M, the middle ordinate. 

M = JCX tan. i D = 50 X 0.0:3055 = 1.528. 

3. GIVEN THE RADIUS R, CHORD C, AND MIDDLE ORDINATE 
M, TO FIND ANY OTHER ORDINATE E K = M', DISTANT d 
FROM N, THE MIDDLE POINT OF THE CHORD. 

K I, = X G ; N K = G L ; E K = E L — N G. 

E L =VG E 2 — X K 2 = VK--fl 2 ; XG (l) = V r R 2 -iC 2 . 
Then EK = M'= \Z~\l 1 — d- — VR 2 — i C*. 

4. It is a property of the parabola, that ordinates vary as the 
products of their abscissas. This property may be assigned to 
the circle in cases where the arc encloses a small angle. 
Applying it here we have — 

HX:EK::AXXXB:AKXKB. 

Call any segments AK, K B, of the chord, a and b. 

Then M : M' :: £ C 2 : ab, ,\ M' = M X 4 ab -f- C 2 . 

Example. 
M = 1.528, C = 100, a = GO, b = 40; to find W. 

M' === 1.528 X 9600 -f- 10000 = 1.528 X 0.96 = 1.467. 

5. Multiply the corresponding ordinate of a 1° curve from 
the annexed table by the degree of curvature: the product will 

1 be the ordinate sought. 



OUDINA TES. 





ORDINATES 


OF A 


1° CURVE, 


CHORD 100 


FEET, 




Distances of the Ordinates from the End of the 100-feet Chord. 


Middle 
Feet. 

50 


Feet. 
45 


Feet. 
40 


Feet. 
35 


Feet. 
30 


Feet. 
25 


Feet. 

20 


Feet. 
15 


Feet. 
10 


Feet. 
5 


Lengths of the Ordinates in Feet. 


.218 


.216 


.209 


.193 


.183 


.164 


.140 


.111 


.078 


.041 























Example, 

What is the ordinate of a G° curve, 30 feet from the end of 
the 100-feet chord? 

The corresponding tabular ordinate of a 1° curve is .183; 
which, multiplied by 6, gives 1.098, the required ordinate. 

6. A quick way of laying off ordinates on the ground, and 
one sufficiently exact for the field, is, after fixing the point II 
by means of the middle ordinate II X, to stretch a line from 
II to A, and make the middle ordinate F O = \ II N; then from 
F to A and F to H, making the middle ordinates = |FO; and 
so on. 

7. A good track-layer will seldom require points at shorter 
intervals than 25 feet. 



TEACIXG CUEYES 

AND 

TUENING OBSTACLES IN THE FIELD. 
XX.— XXIII. 



TRACING CURVES AND TURNING 
OBSTACLES IN THE FIELD. 



XX. 

TO TRACE A CURVE OX TITE GROUND WITH TIIE 
CHAIN ONLY. 

1. This is best laugh t by an example. 




Example, 
From a point B, IS feet in advance of A, on tangent A B, to 
trace a curve of 367 feet radius to the right, with chords 66 feet 
long, and consuming an angle of 34° 21'. 

63 



64 TO TRACE A CURVE OK THE GROUND. 

2. First, dividing half the unit chord, or 33 feet, by the 
radius, 367 feet (XVIII., 18), we have 0.09+ for the sine of the 
tangential angle, corresponding to an angle of 5° 10' : the de- 
flection angle, therefore, is 10° 20'. The tangential distance 
corresponding to the angle 5° 10', and chord 66 feet, is equa 
(XVIII., 22) to twice the chord multiplied by the sine of half 
the tangential angle, = 132 X 0.04507 == 5.95 feet. The deflec 
tion distance (XVIII., 19) is equal to twice the chord multi 
plied by the sine of half the deflection angle, = 132 X 0.09-f 
= 11.88, say 11.9 feet. 

3. To find the length of the curve (XVIII., 13): Divide the 
total central angle by the degree of curvature. The central 
angle, 34° 27', is equivalent to 2067 minutes; dividing by 
620, the number of minutes in the deflection angle, we have 
3.33, the number of chord lengths in the curve, = 3J chains = 
220 feet. 

If A be a stake numbered 2, then the point of curvature, B, 
will be 2.18, and- the point of tangent, F, will fall at 2.18 + 
3.22 = stake 5.40. 

4. To determine the tangential distance C P, to the firs 
stake on the curve, either of two methods may be used : — 

First, The sine of any tangential angle is equal to half the 
chord which limits the angle on one side divided by radius. 
The limiting chord BC in this instance is equal to 66 — 18 = 
48 feet; half of 48, therefore, or 24 feet, divided by radius, 367 
feet, gives 0.0654, the sine of 3° 45' = tangential angle PBC. 
The sine of half this angle multiplied by twice the given chord 
= 0.0327 X 96 = 3.14 feet, the tangential distance C P. 

5. Secondly, CP may be found as follows, assuming that 

the functions of small 
\e angles vary directly as 

the angles themselves, 
and vice versa. 

Let BF be a portion 
of the curve. Make the 
tangent B E equal to the 
chord B F, and strike the 
arc E F. Draw the sub- 
chord B C, and strike the 
arc C P. Prolong B C to D. E F may be taken as the tangen- 
tial distance due to the whole chord BF, and PC the tangen- 
tial distance due to the sub-chord BC. 



1 




TO TRACE A CURVE ON THE GROUND. 65 

Then PC :ED::BC:BD or BF; and, by the foregoing 
supposition, ED:EF::BC:BF. Combining these propor- 
tions, and cancelling ED, we have PC : EF : : BC 2 : BF 2 ,\ 
PC=EF X (BC-fBF) 2 . 

In words, the tangential distance for a sub-chord is to that 
for a whole chord as the square of the sub-chord is to the 
square of the whole chord. The same is true of deflection dis- 
tances. 

6. In the example we are treating, the tangential distance for 
the whole chord of 66 feet has been found to be 5.95 feet; 
that for 48 feet, therefore, is 5.95 X 48 2 -i- 66 2 = 5.95 X 0.528 
= 3.14, as before. 

Stretch the 48 feet of chain from B to P, in prolongation of 
tangent A B, and mark the point P ; then step aside, and stretch 
from B to C, making the distance PC = 3.14 feet: C will be a 
stake on the curve. 

7. Next, run out the whole chain length from C to O in the 
range B C. To find O D, suppose the line N C T to be drawn 
tangent to the curve at C. Then N" D may be considered the 
tangential distance due to the whole chord, =5.95, as above 
determined. 

The angle O C N = T C B=P B C (XVI., 4); and (5) 

ON:ND::BC:CD.*.ON = NDxBC-fCD, ie.;OD 
= N D + O N = N D + N D X^B C-fC D)= 5.95 X 1 +(48 
-5- 66) = 5.95 X 1.727 = 10.27. 

8. The point N may be fixed otherwise by laying off B T = 
C P, and running out the chain length C N in the range C T. 
The point D on the curve may then be fixed by making N" D 
equal to 5.95 feet, the tangential distance. 

Next run out the chain to M, in the range C D ; make M E 
equal to the deflection distance, 11.9 feet, and fix the point E. 
The points C, D, and E will be stakes 3, 4, and 5 on the curve. 

1* . To set the point of tangent, F, at stake 5.40, prolong the 
ch\\\\\ line D E for 40 feet to L, and suppose YE to be drawn 
tangent to the curve at E. Then the angle L E Y is equal to 
the tangential angle of the curve ; and the sub-tangential dis- 
tance L Y is to the whole tangential distance due to the 66- 
feet chord, as the sub-chord is to the whole chord (5); i.e., 
LY = 5.95 X 40 -f- 66 = 3.6 feet. 

By the method illustrated in (6), the distance FY will bo 



66 TO TRACE A CURVE ON THE GROUND. 

equal to 5.95 X 40 2 -f- 66 2 = 5.95 X 0.367 = 2.18 feet. Wi* 
the distance LF = 3.6 + 2.18 = 5.78 feet, thus obtained, and 
the sub-chord E F = 40 feet, the point of tangent F may be 
established. 

10. Next, set off UE = FY = 2.18 feet, and lay out FK in 
prolongation of the range UF; FK wi 1 ! be in the line of the 
terminal tangent. 

11. This analysis has been somewhat minute and detailed. 
in order that the subject may be thoroughly understood. An 
instrument for measuring angles should always be used in rail- 
road service: it greatly simplifies and abridges the labor of 
tracing field-curves, and gives more exact results. But occa- 
sions sometimes rise, in miscellaneous practice, when strict 
accuracy is not required, and the chain only can be had : the 
young engineer should qualify against such occasions. 



XXI. 

TO TRACE A CURVE ON THE GROUND WITH 
TRANSIT AND 100-FEET CHAIN. 

1. This, also, is best taught by an example. 

Let it be a general rule, in locating, to fix the intersection of 
tangents, and to set the tangent points, or the P. C. at least, 
from the P. I. There are exceptional conditions, as a steep 
hillside, timber or broken ground, a very long arc, unimpor- 
tance of exact conformity to the project, and the like, which 
warrant its omission; but where these conditions do not obtain 
or are not prohibitory, and a snug fit is desirable, time wilj 
usually be saved by fixing the P. I. It often proves serviceable 
as a reference point during construction: on the location, it 
gives confidence in the work and an assurance of safe progress, 
which are well worth a little painstaking beforehand. 

2. Having established the P. I., and found the intersection 
angle to measure, say, 66° 45', the first step is to find the apex 
distances so called, or tangent lengths IB, IF. These are 
each equal to R X tan. i I. If a 7° 30' curve be prescribed to 
close the angle, R X tan. $ I = 704 X 0.659 = 503 feet. 



TO TRACE A CURVE ON THE GROUND. 



67 



Or, referring to Ta- 
ble XVI. , theap. dist. 
corresponding to 66° 
45' is found by inter- 
polation to be 3774.6; 
dividing by 7.5, the 
rate of curvature in 
degrees and decimals, 
we have for the apex 
distance 503 feet, as 
above. 

3. Before disturbing 
the in str anient, which 
is presumed to stand 
in the range of the 
terminal tangent, 
measure I F, = 503 
feet, and set the P. T. 
at F. Then direct the 
telescope to the last 
point fixed on the ini- 
tial range AB, meas- 
ure I B, = 503 feet, 
and set the P. C. at B. 
Move to B. 

4. Suppose the P. C. 
to have fallen at a 
stake 2.50. In order 
to find the length of 
the curve, divide the 
intersection angle by 
the degree of curva- 
ture, having first re- 
duced the minutes in 
each f o liYiiiiliviliM of a 
degree by multiplying 
by 10 and dividing the 
product by 6. Thus 
the intersection angle 
becomes 66.75°, and 
the degree of curva- 
ture 7.5°: dividing the 




68 TO TRACE A CURVE ON THE GROUND. 

former by the latter, we have 890 feet for the length of the 
curve. 

Or, the intersection angle 66° 45' is equivalent to 4005 r , and 
the degree of curvature 7° 30' is equivalent to 450': dividing 
the former by the latter, we have 890 feet for the length of the 
curve, as before. 

5. Adding 8.90 to 2.50, the number of the P. C, the P. T. is 
found to fall at stake 11.40. Let the rear chainman make a 
note of this, that there may be no mistake in the terminal plus. 

6. Next, to determine the proper deflections from the line of 
tangent at B, bear in mind that the deflection for a whole 
chain is half the degree of curvature ; and that, in field-curves 
of more than 300 feet radius, the deflections for sub-chords, 
or plusses, may, without material error, be held to vary directly 
as the sub-chords themselves; that is to say, the sub-deflec- 
tions due to 30, 60, and 80 feet, for instance, will be, to the 
deflection due to 100 feet, as 30, 60, and 80 are to 100. 

7. Thus, in the example, 7° 30' being the degree of curva- 
ture, one-half of this, or 3° 45 r , will be the deflection due to a 
chord of 100 feet; and -ffa of this, or a deflection of 1° 52i' 
from the line of tangent at B, will fix stake 3, 50 feet distant 
on the curve. 

8. The following is a, simple rule for finding sub-deflec- 
tions : — 

Multiply the sub-chord in feet by the rate of curvature in 
degrees and decimals : three-tenths of the product will be the 
sub-deflection in minutes. 

Thus, in the example, 50 X 7.5 = 375, and 375 X 0.3 — 
112.5' == 1° 52£', as before. 

9. Having set stake 3, stakes 4 and 5 will be fixed by succes- 
sive deflections of 3° 45'. In establishing stake 5, the index 
will *-ead, 1° 52^ + 3° 45' + 3° 45' = 9° 22f = angle CBS. 

10. Suppose the instrument moved to 5. See that the ver- 
nier has not been disturbed, backsight to B, and deflect 9° 22|' 
right; i.e., double the index angle. The index will now read, 
18 r 45 ; = the angle 1CD; and the telescope will be directed 
along the line C D, tangent to the curve at 5, for the reason 
that the angle B5C has been made equal to the angle CB5 
(XVI. 4). 

Proceed with successive deflections of 3° 45 r from this tan- 
gent, and set stakes 6, 7, 8, and 9, at intervals of 100 feet. 

11. Suppose the instrument moved to 9. In fixing this 



TO TRACE A CURVE ON THE GROUND. 69 

stake, the index will read, 18° 45' -f- 4 times the constant angle 
30 45', = 18° 45' + 15° = angle I CD + angle D5 9, =33° 
45'. In order to place the telescope in the line DE, tangent to 
the curve at 9, it is now necessary to turn an angle to the 
right, from backsight to 5, equal to D95 = D59 = 15°; i.e., 
the vernier should be moved from 33° 45' to 33° 45' -f- 15° = 
48° 45'. The telescope will then be in tangent at 9. 

12. A simple rule for finding the index angle which shall 
place the telescope in tangent at any point on the curve is as 
follows: — 

From double the index angle which fixed the given point, sub- 
tract the index reading in tangent at the last turning-point : the 
remainder will be the required index angle. 

Thus the index angle which established stake 9 was 33° 
45'. Double this angle will be 67° 30'; subtracting 18° 45', the 
reading in tangent at the last turning-point, we have 48° 45', 
the required index angle, as before. 

The reasons for the rule will be obvious from an examina- 
tion of the figure. 

13. Being in tangent, then, at 9, and the index reading 48° 
45', a deflection of 3° 45 7 will fix 10: a further deflection of 3° 
45' will fix 11, and the index will stand at 48° 45' + 7° 30' = 
56° 15'. 

14. To find the deflection corresponding to the sub-chord 11 
F, =40 feet: by the foregoing rule (8), the degree of curva- 
ture, 7.5, multiplied by 40, the length of the sub-chord in feet, 
gives a product of 300, three-tenths of which amount to 90 
minutes = 1° 30'. Adding 1° 30' to 56° 15', makes the index 
angle 57° 45' to fix the P. T. at 11.40. 

15. Move to the P. T. at 11.40, see that the vernier has not 
been disturbed, and backsight to 9. By the foregoing rule 
(12), double the index angle, 57° 45', less the angle in tangent 
at 9, the last turning-point, 48° 45', = 115° 30' — 48° 45', = 
()0° 45', = the index angle in tangent at the P. T., = the tota] 
angle consumed by the curve. The work thus proves itself. 

16. The preceding example would appear in the field-book 
as follows: — 



70 



TO TRACE A CURVE ON THE GROUND. 





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TO TRACE A CURVE ON THE GROUXb. 



7i 



17. This mode of running curves secures a record of each 
step in the proceeding; so that, if any error occurs, it can 
readily he detected. At each turning-point, the number in 
the " tangent" column must correspond with the central angle 
due to the length of curve to that point; and at the P. T. that 
number must correspond with the total central angle. The 
work can thus be checked with facility during its progress, 
and checks itself at the end. 

18. The young transitman is recommended to rule blanks 
after the pattern given, and exercise himself thoroughly in 
computing the parts, and recording the field-notes of various 
curves assumed at will : drawings are not necessary. 

19. Another method, and in some respects a better one, 
is, before starting on a curve or during its progress (prefer- 

* ably the first), 
to record for 
all stations, the 
deflect ions 
which would 
locate them if the instru- 
ment remained at P. C. 
Obviously the final deflec- 
tion thus recorded would 
be 4ialf the central angle of the 
w T hole curve; and, if the instru- 
ment was placed anyw r here on the 
curve, a B. S. to P. C. and deflec- 
tion of half that central angle w^ould lo- 
cate the P. T. The author is indebted 
to Mr. Robert Burgess, C.E., for recom- 
mending this method. However, Mr. 
Daniel J. Hauer has elucidated it by the 
following example, copied from that given 
on page 67, but a new figure is used. 

20. Special attention is called to the use 
of Table XVI, namely, table of chords, 
middle ordinates, external secants, and 
apex distances of a 1° curve. In locating 
and running railroad curves, nearly all 
field calculations can be avoided by the use of this table. 
This alone prevents many errors, as a transitman is frequently 




72 



TO TRACE A CURVE ON THE GROUND. 



bothered by the questioning of his assistants and finds it 
difficult to work formulse under a gleaming hot sun or with 
fingers stiff from the cold. It is advisable to find and set 
down in the field book all the notes necessary to run the 
curve before beginning the work, so that all figures can be 
checked before setting any stakes. Time is saved in so doing. 
Table XVI saves all field calculations, except simple division. 
Thus, referring to Table XVI, the tangent or Apex Dist. cor- 
responding to 66° 45' is found by interpolation to be 3774.6 
feet; dividing by 7.5, the rate of curvature in degrees and 
decimals, we have the Apex Dist., 503 feet. The length of 
curve is found as shown in Paragraph 4, p. 67. The total 
angle, the Apex. Dist., length of curve, and the notation P. I. 
set or not set, are put down in the field book. Then the 
deflection angles are listed, being calculated as set forth on 
p. 68. These notes are all shown as kept in an ordinary field 
book and give another method of recording transit notes. 
Notice that 50-foot stations are given. This is common to- 
day, as better work is done when curves are so staked. On 
very sharp curves sometimes 25-f oot stations are found neces- 
sary. Such stakes are found useful in cross-sectioning, for 
grading and also track-work. It is not necessary to wait un- 
til all the deflections are calculated to see if they check by 
doubling the total deflection angle, which should equal the 
angle of intersection; for they can be checked as they are set 







Def. 


Total 


Calc. 


Mag. 


Re- 


Sta. 




Angle 


Angle. 


Curve. 


Curve. 


marks. 


12 


o / 


o / 


o / 


o 


o / 




+40 


O P.T. 


33 22± 


\ 66 45 


N. 15 E. 


N. 15 10 E. 


P.I. set 


11 




31 52§ 








T or 


+50 




30 


. 9° = A=18° 






apex 


10 




28 07£ 






dist. = 


+50 




26 15 








503 


9 


O 


24 22i 










+50 




22 30 








A = 


8 




20 37£ 








66° 45' 


+ 50 




18 45 










7 




16 52± 


. 15°=A=30° 






Log C 


+50 




15 






=890. 


6 




13 07| 










+50 




11 15 










5 


o 


9 22^ 










+50 




7 30 










4 




5 37£ 










+50 




3 45 










3 


o / 


1 52* 










+50 


OP.C.7 30R. 













2 













TO TRACE A CURVE ON THE GROUND. 72a 

down. Thus, every 200 feet the deflection angle added, 
equals the degree of curve. Then, too, the deflections 4+50 
is double that for 3 + 50, that for 6+50 is double that 
for 4 + 50, and others are checked in the same manner. 

21. After these notes are all recorded the transit is placed 
at P. C, Sta. 2+50 with the vernier at zero; B. S. is taken 
on the hub on tangent and the stations are all set in order 
to 5; beyond which, owing to rolling ground, they carmot be 
seen. The instrument is then moved to 5; and with vernier 
set at zero a B. S. is taken on the P. C. A deflection of 
9° 22 §' brings the instrument on tangent C. D., and a de- 
flection of 11° 15' sets 5 + 50, and so on, using the recorded 
deflections. Another set up is found necessary at Sta. 9. 
Moving there the transitman observes the general rule applic- 
able to this method, namely to set vernier at each new instru- 
ment point y to the angle recorded opposite the point of his 
proposed backsight. In this case that point is Sta. 5. He 
therefore sets the vernier at 9° 22 J ', B. S. to 5, then deflection 
of 24° 22f, places the telescope tangent to Sta. 9. Then from 
the recorded deflections the rest of the stations are set and 
the P. T. placed. Placing the transit on the P. T. taking 
a B. S. on 9, vernier set at 24° 22J', and a final deflection 
of 32° 22^', turns off the tangent ahead. 

22. This method, in connection with the Table XVI, is 
also a simple manner of passing obstructions on curves. 
Thus with a house located at Sta. 7, see figure, it can be passed 
as follows. The point D comes within the house, but the long 
chord from 5 to 9 goes to the right of the house. The deflec- 
tion from Sta. 5 to Sta, 9 is 24° 22J' less 9° 22J or 15°. Thus 
the intersection angle at D is 30°. Now turn to Table XVI, 
and the long chord for an angle of 30° is 2966.1 feet. Divide 
this by 7.5, degree of curvature and the chord between 
Sta. 5 and Sta. 9 is found to be 395.5 feet. Now from 5 the 
stations to 6 + 50 can be set. Then by deflecting 24° 22J', 
and measuring the long chord 395.5', Sta. 9 is set, The tran- 
sit being placed there, the stakes back to 7 + 50 are put in, 
and the house is located for plotting. Then by a B. S. to 
5 the rest of the curve can be located as previously described. 

If, however, another building should obstruct the line 
between Sta. 10 and 10 + 50 and the long chord cannot be 
used, it can be passed as follows. The deflection angle 



72 7 j TO TRACE A CURVE ON THE GROUND. 

between Sta. 9 and the P. T. is 9°, thus the intersection angle 
at E is 18°. Now by Table XVI the Apex Dist. for 18° is 
found to be 907.52 feet. Divide this by 7.5, degree of cur- 
vature, and the Apex Dist. for this art of the curve is found 
to be 121 feet. Now with the transit set on 9, and vernier 
9° 22|' with a B. S. on 5, a deflection of 24° 22J', places the 
instrument tangent at 9, on line D. E. and 121 feet measured 
from 9, sets E. From 9, Stas. 9 + 50 and 10 are set. From 
E with a B. S. at 9, an angle of 18° is turned and by measuring 
121 feet, F. or the P. T., is set. Then set up the transit 
on the P. T. with the vernier set at 33° 22§' with the tele- 
scope looking ahead towards 12. Plunge the telescope over 
and B. S. on E, then by using the recorded deflections, the 
stakes at 11 and 10 + 50 can be placed. In like manner if 
the P. T. should come in a body of water, as a river or lake, 
the P. I. at E can be set and triangulation across the water 
can be made from E. 

23. Table XVI also gives the middle ordinate and external 
secant of a 1° curve. These, too, are found useful in locating 
curves, as the middle ordinate and external secant can be found 
for the whole curve or for any part of it in the same manner as 
the long chord or Apex Dist. With the point I located, by 
means of a pocket compass the approximate intersections 
angle can be obtained, and then by Table XVI the external 
secant for any degree of curve is mentally calculated. The 
center point of the curve can be found with a tape line cr 
even by stepping off the distance. Thus the chief of the party 
can decide upon the curve in advance, saving much time and 
field work, especially if a hand level is used. Table XVI is 
thus found to be a useful one for curves. 

24 The chief advantages of this method of tracing a curve 
on the ground are: the completeness of the record for sub- 
sequent use, as, once calculated, it can be used even after the 
road is in operation; its adaptation to a retracing of the lire 
backward as well as forward; and also to picking up a curve 
in the middle and running it either way. Observe particu- 
larly the slight labor this method imposes, especially when 
Table XVI is used, in mental arithmetic, its great simplicity 
permitting any one who can read figures and turn an angle 
to relieve the transitman on occasion. 



TURNING OBSTACLES TO VISION IN TANGENT. 73 

XXII. 
TURNING OBSTACLES TO VISION IN TANGENT. 



Let lines BC, CE, FG, cut 



1. Draw CF parallel to AB. 
these parallels at equal 
inclinations. Call this 
angle I. Then B C = 
CE = FG. BE = 
BD + DE = 2BD. 
But BD = BC cos. I, .-. BE = 2 BC cos. I. 
BG = EG + BE = CF + 2BC cos. I. 




E G = C F. 



Example. 

Suppose B to be a stake 24.50 on the tangent AB, and that 
a deflection left of 10° be made there for 200 feet to a point C. 
Set transit at C, vernier reading 10° left. B S to B, and deflect 
20° right. Yernier will now read 10° right, and telescope will 
be in line C E. Make C E = 200 feet. Move to E. See that 
vernier still reads 10° right. BS to C, and turn 10° left. Ver- 
nier will now read zero, and telescope will be in line E G, or 
tangent AB prolonged. 

Distance BE = 2 B C cos. 1 = 2 (200 cos. 10°) =400 X .985 
= 394 feet. Then E == 24.50 -t 394, = stake 28.44 on tangent 
A B prolonged. 

If a parallel line C F were run, a deflection of 10° right would 
be made at each of the points C and F. If C F were 250 feet, 
then B G would be = 250 + 394 = 644 feet, and the point G 
would fall at stake 30.94 on tangent A B prolonged. 

2. If angle I = 60°, the other conditions of above method 
being observed, triangle BHE will be equilateral, and BE = 

B H = H E. If the parallel D C 
or DF be run, BE = BD -f 
DC, and BG = BD + DF. 
For field work see last example. 
3. In turning obstacles by 
either of these methods, the 
angles should be measured with extreme niceness. Handle 
the instrument lightly, to avoid jarring the vernier; and, if 
possible, observe well-defined distant objects in the several 
short ranges, that the lines of foresight and backsight may 
accurately coincide. 




74 TURNING OBSTACLES TO MEASUREMENT IN TANGENT. 

In locating, the following method is preferable to those given 
above, and should always be used on long tangents. 

4. Having established points A and B on the centre line, 
the farther apart the better within limits of distinct vision, 
set off the equal 




rectangular d i s - 

tances AE, B F, 

ranging clear of 

the obstacle. 

Place the transit at E or F, fix points G and H on the forward 

range, and, rectangularly to these points, establish others on 

the forward range of the centre line at C and D. The offset 

distances should be measured very carefully with the rod, or 

with a steel tape if they exceed in length the pocket rule 

which every engineer should have about him. 



XXIII. 




TURNING OBSTACLES TO MEASUREMENT IN 
TANGENT. 

1. Fix a point on tangent A B prolonged at E. Lay off at B 
a perpendicular of any convenient length. Move the instru- 
ment to D, make the angle B D A 
= B D E, and mark the point of 
intersection A. By reason of 
symmetry in the triangles AD B, 
B D E, A B = B E, and may be 
measured on the ground. 

2. Or, fix the point E, and lay 
off the perpendicular BD as before. Move to D, direct the 
telescope to E, turn a right angle E D C, mark the point of 
intersection C, and measure CB. Then, by reason of simi* 
larity in the triangles C BD, DBE, CB : BD :: BD : BE, 
.-. BE = BD 2 ~BC. 

Example. 
Suppose BD to be 60 feet, and B C 40 feet. Then BD 2 -f 
BC = 3600 -f- 40 = 90 feet = BE. 

3. Or, with the instrument at D, measure the angle BDK 
ThenBE = BD tan. BDE. 



TURNING OBSTACLES TO MEASUREMENT IN TANGENT 



BD tan. BDE = 



Example. 
B D = 120 feet angle B D E = 54° 40'. 
120 X 1.41 = 169.2 feet = BE. 
4. Or, without an instrument, lay off any convenient lines B F 

or C H. Mark the middle point 
D. Line out H G, parallel to 
AB. Mark on it the point G 
in range with D and E. Then 
GF = BE, orGH = CE. 

5. Should the use of a right 
angle be inconvenient, turn any 
angle EBD = x, measure BD 
about equal by estimation to B E, if the ground permits, and 
set a point D. Move to D, and measure angle B D E => y. 

Then the angle B E D, or z, = 180 — (x -f- y), and, by trigonom- 
etry, sin. z : sin. y :: BD : BE,.*. BE = BD sin. y -f- sin. z. 




Let x = 44° 02', y ■■ 



Example. 
71° 48', BD = 300 feet. 



Then z = 180° — (x + y) = 180° — (44° 02' + 71° 48') = 
180° — 115° 50' = G4° 10'. BE = BD sin. y -f- sin. z = 300 
sin. 71° 48' ~ sin. 64° 10' = 300 X .95 -f- .90 = 316.6 feet. 

The calculation by logarithms would be as follows: — 

Log. 300 2.477121 

Log. sin. 71° 48' 9.977711 

Sum . 12.454832 

Log. sin. 64° 10' . . ; 9.954274 

Log. 316.6. Diff 2.500558 



If E is invisible from B, extend the line D B towards C, 
until a line C E clears the obstacle. The point E must then 
be established by intersection of the sides C E, D E, in triangle 
CDE. Supposing the extension BC to have been 120 feet, 
the side CD will be 420 feet, the angle y 71° 48'; and, by a 
calculation similar to the above, the side DE, opposite angle x 
in the lesser triangle, identical with DE in the larger one, will 
be found to be 231.7 feet. The sum of the angles at the base 
C E of the triangle C D E = 180° — y = 180° — 71° 4S' = 10S° 
12' By trigonometry, two sides and the included angle being 



76 TURNING OBSTACLES TO MEASUREMENT IN TANGENT. 

known in any plane triangle, the sum of the knQwn sides is to 
their difference as the tangent of the half sum of angles at base 
is to the tangent of half their difference. In triangle CDE, 
therefore, CD + DE or 651.7 : CD — DE or 188.3 :: tan. 
108.12 -f 2or tan. 54° 06' : tan. 54° 06' X 188.3 ~ 651.7 = 
.399 = tan. 21° 45', = half the difference of the angles at the 
base. 

Log. 188.3 2.274850 

Tan. 54° 06' 0.140334 

Sum 2.415184 

Log. 651.7 2.814048 

Tan. 21° 45'. Diff 9.601136 

The angle at C, being evidently the lesser of the two angles 
at the base, is equal to the half sum of these angles decreased 
by their half difference, = 54° 06' — 21° 45' = 32° 21'. 

Set the transit, then, at C, foresight to D, deflect 32° 21' left, 
and fix in that range two points F and G, between which a 
cord may be stretched, and as nearly as can be judged on 
opposite sides of E. Move to D, foresight to C, deflect 71° 48' 
right, and establish a point E at intersection with F G. Cross 
to E, BS to D, and deflect the angle z = 64° 10' into the line 
of the tangent A B prolonged. 



SUGGESTION'S AS TO MELD -WORK 
AND LOCATION -PROJECTS. 

XXIV. -XXV. 



SUGGESTIONS AS TO FIELD-WORK 
AND LOCATION -PROJECTS. 



XXIV. 

SUGGESTIONS CONCERNING FIELD-WORK. 

1. The Chief Engineer, after conference with his em- 
ployers in regard to the character of the work contemplated, 
and its general route, should, before organizing field-corps, go 
over the ground in both directions, and, aided by the best 
attainable maps, qualify himself by actual observation to in- 
struct his assistants as to the conduct of the survey. Equipped 
with hand-level, pocket-compass, and in rough regions with 
the -aneroid, he can often not only prescribe lines for examina- 
tion, but indicate the gradients to be tried, thus saving a vast 
amount of random labor and needless expense. Such thorough 
preliminary exploration is due-both to himself and his princi- 
pals: it is too often omitted, or done with a perfunctory rush. 
In broken topography, no maps, notes, or information derived 
from others can supply the want of personal acquaintance with 
the ground itself. He must indispensably make that acquaint- 
ance, in order to project an intelligent location, — a work which 
should rarely be delegated; being capital service, it comes 
within the special function of the chief engineer, and only the 
necessary distribution of labor attending a great charge should 
relieve him from its direct performance. 

2. A Field-Corps in settled regions generally consists of 
one senior assistant or chief of corps, one transitman, one 
leveller, one rodman, two chainmen, one slopeman, and two 
or more axemen. 

The following notes in regard to the allotment of duties and 
the conduct of work may be acceptable. They are copied 
from the writer's memoranda for the guidance of his field- 
parties, with the addition of some detail, and practical hints 
here and there, to aid the inexperienced. 

79 



80 SUGGESTIONS CONCERNING FIELD-WORK. 

3. The Senior Assistant will receive instructions from 
the principal assistant in charge, or the chief engineer, and 
will act exclusively under his direction. 

He will be held responsible for the good conduct of the corps, 
and for the rapid, exact, and economical performance of the 
work. Indecent or blasphemous outcries in the field should 
be prohibited. The writer's various travel by land and sea 
lias brought him acquainted with many cultivated, estimable, 
energetic, profane fellows, but not one in whom swearing was 
\ grace ; nor has he ever seen an instance where it forwarded 
Work. Those considerate of others' pride and self-respect 
Vvill generally find that a good leader makes good followers. 

The senior assistant is empowered to appoint and dismiss 
employes below the rank of rodman, and will report any 
inefficiency or neglect of duty in the ranks above . to his 
chief. 

He will pay the authorized expenses of the corps for sup- 
plies, repairs, transportation, and subsistence, taking duplicate 
vouchers. Accommodations should be sought near the work. 
When not thus obtainable, transportation to and from the 
field is to be regarded as a measure of economy for the com- 
pany, compensating the expense incurred by saving time and 
labor. 

He will superintend field operations in person, keeping in 
advance of the transit to direct and expedite the work, and 
establish the turning-points. On preliminary surveys, the axe 
should be little used ; and on alternative locations, or such as 
may be subject to revision, trees over four inches in diameter 
need rarely be felled. 

He should be patient with sensitive landholders. He will 
find exercise for that amiable virtue, also, with the field vis- 
itors who so often spare time from useful toil to tell him he is 
on the wrong line, and to show him where the right one is. 

Note for record the kind and quality of material to be moved, 
observing quarries, wells, or other indications for the purpose; 
the timber and rock in the country traversed, with a view to 
their use in construction, and the widths of passage to be pro- 
vided for streams, together with the character of their banks 
and beds. 

Note the names of residents in the immediate vicinity of the 
work on survey; and, on location, cause the property-lines to 
be observed and recorded also when convenient. 



SUGGESTIONS COKCERKWG FIELD-WOtiK. 



81 



Always begin grade-lines at the summit, and work down. 
For such service, carry habitually a slip of profile paper, say 
six inches wide and two feet long. Rule the proposed grade- 
line on it, assume a summit cut, mark the stations, and start 
down. When at fault, the elevation can be spotted on the 
profile, which will show at a glance, without any calculation, 
how you stand in relation to grade. 

The work of each day should be compiled and recorded in 
the evening, that no delay may result from the loss or deface- 
ment of a field-book. 

FORM FOR SURVEY RECORD. 



Sta. 


Dis. 


Deflec. 


Course 


M.C. 


Eleva. 


Slope. 


1 
Remarks. 



















FORM FOR LOCATIOX RECORD. 



o 

H 

t 


a 
o 

CO 

5 


co 
O 

O 


w 

CO 

K 

P 
O 

O 

6 
< 


© 

< 
> 


a 

< 

O 


R 


i, 
o 

< 

< 
> 


q 

5 


Remarks. 



























On location, check the transitman's calculation of the length 
of each curve and the fractional deflections. 

The senior assistant must be qualified to locate a line accu- 
rately on the ground from the project furnished him. Lateral 
deviations exceeding five feet on ten-degree slopes, three feet 
on fifteen-degree slopes, and two feet on twenty-degree slopes, 
will be considered errors requiring correction. Measurements 
to the experimental line should be made and noted frequently, 
in order not only to check the field-work, but that the line 
may by means of them be laid down on the map. 

The senior assistant will supply himself with drawing in- 
struments, colors, brushes, and the like personal furniture of 
an engineer. He will take care also that the stationery, field- 
books, instruments, and other articles of outfit supplied by the 
company, are not misused. His field equipment should always 
include a hand-level and a pocket-compass: to these may be 



82 SUGGESTIONS CONCERNING FIELD-WORK. 



ited: 



added a straight, round staff, five or six feet long, steel pointed \ 
it will be found exceedingly useful. 

If without a topographer, he should make sketches of irregu* 
lar ground, of streams, buildings, roads, and the like, to help 
in compiling the map. 

In hilly or wooded districts, the front chainman carries the 
flag on survey, and is at the head of the line. In open, plain 
country, work is greatly forwarded by detaching an axeman 
with flag, to accompany the senior in advance, and set turning- 
points for the transit. The transitman follows as rapidly as 
possible, and the chainmen come after, lining in their stakes 
by the eye from point to point. The whole force is thus kept 
pretty steadily in motion. 

On wide plains, a set of chain-pins may be used, and survey- 
stakes placed five hundred or a thousand feet asunder. Very 
often stakes at intervals of two hundred feet are sufficient, the 
levels being taken every hundred. Location stakes are put in 
every hundred feet. 

4. The Transitman will be expected to keep his instru 
ment in adjustment, and to be quick and accurate in its manip 
ulation. It is not needful to plant it as if for eternity. Oi, 
the contrary, it should be set gently, the legs thrust but slight 1; 
into the ground, and the screws worked without straining. 

On long tangents it is a good plan to reverse the instrument 
at each new point, putting the north and south ends forward 
alternately. Small errors in adjustment are thus balanced it 
some measure. Select also, in such a case, some distant objeel 
in range, when practicable, to run by. The telescope, in win* 
or sun, will sometimes warp a little out of line. 

iSever omit to note both the calculated and magnetic bear 
ings of the lines on survey, and of the tangents on location. 
Guard against the error of reading deflections or bearings from 
the wrong ten mark; as, for instance, 34 instead of 26. 

At the beginning of a curve, let the rear chainman know the 
plus of the P. T. Tell the front chainman the degree of curve, 
and instruct him how, by multiplying 1.75 by the degree, he 
can find the distance of each full station from the range of the 
last two. A quick fellow will soon pick this up, and become 
wonderfully skilful in practice. Thus accomplished, he is a 
check on wrong deflections. 

In running curves, a tangential angle of fifteen degrees from 
one point should seldom be exceeded : twenty degrees is to be 
regarded as a maximum. 



SUGGESTIONS CONCERNING FIELD-WORK. 83 

Carry a pocket-compass, and observe with it the magnetic 
bearings of streams and roads crossed. 

Kecord daily each day's run; fill out the distance column, 
transcribe the chain-book, and, on location, record the apex 
distances also in the column of remarks. 

On survey, do not erase from the field-book the notes of 
abandoned lines. Simply cancel, and mark them " aban* 
doned," in such manner that they may still be legible. 

When required by the senior assistant, the transitman will 
aid in the making of maps. 

5. The Leveller must be familiar with the adjustments 
of his instrument, keep it in order, and handle it rapidly. 

On survey, establish and mark benches at half-mile intervals; 
on location, four to the mile when practicable. 

Note the surface elevations, the depths, and the flood heights, 
of all considerable streams crossed. Take elevations in the 
beds of small streams. 

Six hundred feet each way should be regarded as the maxi- 
mum sweep of the level. 

Carry a hand-level, and thus save the time required to peg 
across narrow hollows, or over heights which can be turned 
with the instrument. 

The leveller should record his work, and make up the profile 
daily. 

6. The Kodman will give his-intermediates close by the sta- 
tions, observing the number of each one as a check on the 
chainmen, and calling it out to the leveller. He should have 
an eye to abrupt irregularities in the ground, and give plus 
elevations when necessary. 

He will keep note of bench-marks and turning-pegs, describ- 
ing the latter occasionally with reference to the nearest stake, 
that the levels may be taken up speedily in case of a revision 
of the line. 

When unaccompanied by an axeman, the rodman is equipped 
with belt and hatchet. Sometimes he is furnished also with a 
steel pin for turning on. The pin has a ring through the head, 
by which it may be hung to a spring hook in the belt. 

The rodman will assist the leveller at record and profiles, 
and transcribe the slope-book daily. 

If stakes of survey are set at intervals of two hundred feet 
give rods every hundred feet, as nearly as the midway points 
can be guessed. 



84 THE CURVE-PROTRACTOR. 

7. The Slopeman will give backsights, and take tlie cross 
slopes for one hundred feet on each side of the line at every 
station. 

8. The Rear Chainman will carry a book in which to note 
the turning-points, the crossings of roads, streams, swamps, 
woodland, and, when convenient, property lines also. He will 
hand it in daily to the transitman for record. As each succes- 
sive chain is stretched, the rear chainman calls out the number 
of the stake it is stretched from: this assures the selection of 
the right number for the stake ahead. 

9. One Axeman will be detailed to make stakes, another to 
mark and drive them. Additional axemen may be employed 
at the discretion of the senior assistant, as the work requires 
them. Wanton destruction of timber, fences, growing crops, 
or other property, should not be allowed. Axemen must be 
careful, in passing through the country, to do as little damage 
as possible. 



XXV. 

THE CURVE-PROTRACTOR, AND THE PROJECTING 
OF LOCATIONS. 

1. " The curve-protractor is simply an eight-inch, semi-circu- 
Jar horn protractor, upon which a series of twenty-three curves, 
from half a degree up to eight degrees, is finely engraved, to a 
scale of 400 feet to an inch. After some years' use in his own 
practice, the contrivance was transmitted by the writer to the 
well-known firm of James W. Queen & Co., mathematical- 
instrument makers, New York and Philadelphia, by whom it 
is now manufactured. It greatly facilitates the projecting of 
lines and solution of field-problems on location. It enables 
the engineer, for example, by a short, graphical process and 
rapid inspection, to find the curve which shall close an angle 
between tangents, or terminate a compound curve, and pass at 
the same time through some fixed intermediate point, without 
liability to the errors, and free from the loss of time, involved 
in a tedious calculation. Other applications, such as the nice 
adjustment of line among buildings, on precipitous steeps, and 
the like, wiU suggest themselves to the experience^ reader* 



THE CURVE -PROTRACTOR. 85 

2. For office use, the writer prefers a home-made curve- 
protractor of mica, prepared as follows: Take a thin, clear 
sheet, say six by ten inches, free from bubbles and cracks. 
Block it securely on the drawing-table with thumb-tacks, set- 
ting the shanks close against the edge of the sheet, but not 
piercing it, and the heads lapping its edge. From a centre, 
midway of one of the long sides and near its margin, strike the 
curves from 12° or less, varying outwards by half-degrees to 
G° ; "thence by quarter-degrees to 4° ; and thence by ten-minute 
differences to 2\°. This covers one side of the sheet, the scale 
being 400 feet to an inch. Xow release the sheet, turn it over, 
and on its other face strike the remaining curves, down to 
ten minutes, from centres on the table, in the reverse direc- 
tion, so that they shall cross the first series at a large angle. 
Space them about three-eighths of an inch asunder at the mid- 
dle. Use a needle-point centre for the first series, to avoid 
boring a large hole in the sheet. Add also, on that face, two 
radial lines drawn towards the corners. Score the fractional 
turves very lightly, the full figure curves a little deeper, but 
all of them with steadiness and delicate stress. Practise 
beforehand on a separate slip, for the right intensity of stroke. 
Engrave the numbers with a stiff steel point on the opposite 
side of the sheet to that upon which the corresponding curve 
is traced. Bring the work out by rubbing it with India ink. 
If preferred, the flat curves on the reverse side may be colored 
with carmine. Duplicate protractors will be found useful in 
projecting compound curves. Clip off the four corners, re- 
enforce the edges with a narrow ribbon of tracing-linen, folded 
over them and glued fast, and the article is complete. It is 
perfect for its use; durable, flexible, spotlessly transparent, 
not liable to warp or change dimensions with changes in the 
temperature or moisture of the air, and, withal, takes and pre- 
serves a visible line, thin as the gossamer. 

3. To experienced locating engineers, the curve-protractor 
needs no wordy commendation. Contrasted with the incon- 
venient appliances of the old method, — cardboard, veneer, 
glass, or dividers, — its advantages will be manifest. A few 
hints as to the manner of using it may be in place. 

4. First of all, let the experimental line approximate to the 
probable line of location; and, upon that base, construct a 
contour map, with reference to which special observations 
should be made in the field, and the chaining done with care* 



B6 THE CURVE -PROTRACTOR. 

Extreme accuracy in the contours need not be attempted. 
Note the courses of streams, ravines, and ridges, the average 
slopes at frequent intervals, and, on irregular ground, make 
illustrative sketches to aid in utilizing the other notes. Prac- 
tice gradually teaches how to observe critical points intelli- 
gently, and to record them briefly. In valleys or plains, where 
the location indicated is made up of long tangents and easy 
curves, little detail is required ; but on bluffy, tortuous ground; 
with unavoidable divides to overcome, and long reaches of 
maximum gradient to be fitted, the method by contours is not 
only the simplest and clearest way of compiling necessary 
information, but is an aid to the engineer in projecting the 
right line, which no substitute can fully replace. 

5. The writer is forced by the strong constraint of experi- 
ence to differ on this subject with Mr. Trautwine. The dif- 
ference, however, is a permissible one, and implies no 
lack of grateful respect for that veteran engineer, whose 
books are our handy-books, and to whose genius we are all 
debtors. 

6. Having made the map, with ten-foot contours, suppose, 
for example, that a continuous gradient five miles long is to be 
located. Spread the dividers to 500 feet by the scale, start at 
the foot of the ascent, and step up, complying with the general 
trend of the ground, to the summit. This needful preliminary 
gives about the distance you have to work on, which cannot 
in many cases be derived from the experimental line directly. 
The profile furnishes the height to be overcome; and you are 
thus prepared to assume a summit cut, and determine the 
gradient. Having adopted one, say, of 66 feet per mile, 
observe that this rises five feet in 400 feet. Spread the 
dividers, then, to 400 feet by scale, and stand one leg on or 
near the summit, at a point corresponding to a five or ten unit 
in. the elevation of the gradient. That is to say, if the grade 
elevation at the summit be 362, for instance, stand the leg of 
t\\e dividers a little beyond or a little short of the summit, at a 
point where the grade elevation is 365 or 360. Thence, exer 
cising good judgment to conform in a general way to what the 
location ought to be, and to make no angular indirections which 
cannot be closed with the maximum curvature, step forward 
down the incline. Name each step mentally as it is made, 
355, 350, 345, 340, &c, and spot at the same time with a pencii- 
j>oint the contour or half space, directly opposite, correspond* 



THE CURVE -PROTRACTOR. 87 

ing to it in elevation. Connect the pencil-marks with a faint 
dotted line. 

7. Were the ground a straight, regular hillside, the steps 
would be made directly from contour to half-space, thence to 
the next contour below, and the dotted line would mark out a 
tangent conforming exactly to the ground surface. On devious 
slopes, rounding within the limit of the sharpest permissible 
curve, the same exact conformity could be obtained, if desired, 
and a grade-line laid down which should require the least 
possible expense in building. On irregular, winding ground, 
an approximation only to the dotted line can be made: it is 
levertheless a guide to go by; and, the more nearly the loca- 
/' _>n project approaches it, the lighter will the work of con- 
traction be. The dotted line, in short, is analogous to a 
profile ; and the engineer can prescribe his cuts and fills with 
reference to it, by means of curve or tangent, just as on the 
profile he does the same by means of grade-lines. A fairly 
correct map will enable him to construct a profile from the 
project, and to amend its errors without the trouble and ex- 
pense of tentative field-work. The writer's habitual practice 
has been to base his preliminary estimates on a profile thus 
deduced from the map; and he recommends the practice to 
others. They will be surprised to observe the likeness between 
such a profile, tolerably well done, and that of the subsequent 
location. — 

8. It is a good custom, and one which cannot prudently be 
neglected where long reaches of maximum gradient are en- 
countered, to " slacken grade " on the curves. In making this 
adjustment, the contour map is exceedingly useful. An ap- 
proximate project is first required, in order to determine the 
curvature, and, from that, the varying gradient. The location 
can then be laid down on the map with satisfactory precision. 
Opinions differ as to the right allowance per degree of curva- 
ture, and no experiments seem to have been made from which 
to deduce an authoritative rule. Some say 0.025 per degree 
per 100 feet; others, 0.05; others, variously between the two. 
Probably 0.05 is the safer rate. This amounts to 2.64 feet on a 
mile of continuous one-degree curve, and makes a nine-degree 
curve, about the curve of double resistance at ordinary passen- 
ger speeds. 

9. In projecting locations, the better way generally is to 
strike the curves first. 



THE CURVE -PROTRACTOR. 

10. The following tables may be of assistance. It was need- 
ful, calculating them at all, to calculate them right; but of 
course such exactness as the figures would seem to indicate 
is unattainable in practice. 




11. TABLE SHOWING THE DISTANCE, D, IN FEET, AT WHICH 
A STRAIGHT LINE MUST PASS FROM THE NEAREST 
POINT OF ANY CURVE, STRUCK WITH RADIUS r, IN 
ORDER THAT A TERMINAL BRANCH HAVING RADIUS 
R = 2 r, AND CONSUMING A GIVEN ANGLE, X, MAY 
MERGE IN SAID STRAIGHT LINE. 

D = (.R — r) X (1 — cos. x). 



Angle 






Degree of the Main Curve. 
























X. 


2° 


3° 


4° 


5° 


6° 


7° 


8° 


9° 


10° 


D. 


2° 


1.72 


1.15 


0.86 


0.69 


0.57 


0.49 


0.43 


0.38 


0.34 


3° 


4.01 


2.67 


2.00 


1.60 


1.34 


1.15 


1.00 


0.89 


0.80 


4° 


6.88 


4.58 


3.44 


2.75 


2.29 


1.96 


1.72 


1.53 


1.37 


5° 


10.89 


7.29 


5.44 


4.35 


3.63 


3.11 


2.72 


2.42 


2.18 


6° 


15.76 


10.50 


7.88 


6.30 


5.25 


4.50 


3.94 


3.50 


3.15 


7° 


21.49 


14.32 


10.74 


8.59 


7.16 


6.14 


5.37 


4.77 


4.30 


8° 


28.36 


18.91 


14.18 


11.35 


9.45 


8.10 


7.09 


6.30 


5.67 


9° 


35.24 


23.49 


17.62 


14.09 


11.75 


10.07 


8.81 


7.83 


7.05 


10° 


43.55 


29.13 


21.77 


17.42 


14.52 


12.44 


10.89 


9.68 


8.71 



THE CURVE -PROTRACTOR. 



89 



If R = 1^ r, use half the tabular distance ; if R = 3 r, use 
twice the tabular distance; if R = 4 r, use three times the 
tabular distance, and so on. 




12. TABLE SHOWING THE DISTANCE, d, IN FEET, AT WHICH 
CURVES OF CONTRARY FLEXURE MUST BE PLACED 
ASUNDER IN ORDER THAT THE CONNECTING TANGENT, 
T, MAY BE 300 FEET LONG. 



> 
M 
P 
O 


Degree of Curve. 




> 
P5 






P 




















o 


1° 


2° 


3° 


4° 


5° 


6° . 


7° 


8° 


9° 


10° 


o 


w 

w 
























d. 


1° 


3.9 


5.24 


5.92 


6.29 


6.35T 


6.68 


6.86 


7.00 


7.08 


7.18 


1° 


2° 




7.84 


9.43 


10.38 


11.20 


11.70 


12.20 


12.55 


12.80 


13.06 


2° 


3° 






11.77 


13.43 


14.64 


15.68 


16.45 


17.09 


17.61 


18.05 


3° 


4° 








15.65 


17.39 


18.76 


19.90 


20.82 


21.64 


22.31 


4° 


5° 










19.54 


21.22 


22.76 


24.01 


25.07 


25.97 


5° 


6° 












23.32 


25.20 


26.70 


28.00 


29.13 


6° 


7° 














27.25 


29.01 


30.58 


31.93 


7° 


8° 
















31.05 


32.82 


34.41 


8° 


9° 


















34.82 


36.31 


9° 


10° 




















38.§6 


10. 



Examp les. 

A 7° and 4° should be 19.9 feet asunder; a 5° and 9° should 
be 25.07 feet asunder. 

As approximations, for a connecting tangent 400 feet long, 
take twice the tabular distance: for a connecting tangent 200 
feet long, take half the tabular distance. 



.90 THE CURVE-PROTRACTOR. 

It is thought by some that the parabola is an ideal curve for 
railroads, and should be adopted; by others, that a spiral or 
parabolic " easement, " so called, is sufficient by way of tran- 
sitional flexure from straight line to circular curve; by others, 
that a circular curve compounded with similar terminal curves 
of larger radii is to be preferred; by others, that the circular 
curve alone serves all conditions best. The writer holds with 
the last party up to curves of about 4°, and for sharper ones, 
in the absence of proof to the contrary, believes with the third 
party that circular terminal curves, not less than 200 feet long, 
having half the degree of the main curve, are likeliest to 
meet, in a fair measure, the requirements of actual service. 

Meanwhile the old circular curve continues to do good 
work. On well-regulated lines a curve is usualty indicated to 
the traveller by the inclination of the car only; there is no jar. 
Some years ago one of the most intelligent, experienced, and 
enterprising railroad managements in this country caused a 
thorough practical test to be made of the second device above 
mentioned, with a view to jts adoption if found advisable. 
The engineers and superintendents who made the test reported 
adversely, on experimental grounds. The proposed improve- 
ment was not adopted. 



PROBLEMS IN FIELD LOCATIONS 
XXVL— XXXVII. 



PROBLEMS W FIELD LOCATION. 



XXVI. 

HOW TO PROCEED WHEN THE P. C. IS INACCES- 
SIBLE. 

1. Suppose, for example, a pro- 
jected 5° curve, beginning at stake 
24.20, or B in the diagram. 

First Method. — At any point 
A, which- we will assume to be 
stake 23.40, set up the transit. Let 
it be judged that stake 27, marked 
D m-the diagram, must fall on ac- 
cessible ground. Then the distance 
B D, around the curve is 280 feet, 
corresponding to an angle EJB D of 
7° at the circumference, or an angle 
of 14° at the centre. The chord of 
a 1° curve consuming this angle, by 
Table XVI., is 1,396.6 feet; that of 
a 5° curve, B D in the figure, is one-fifth of this, or 279.3 feet. 
In the triangle A B D are thus known the sides A B, B D, and 
the sum of the angles at A and D, which sum is equal to the 
angle E B D. 

Hence, by trigonometry, — 

As the sum of the sides given = 359.3 AC .... 7.444543 

Is to their difference =199.3 2.299507 

So is tan. h sum of angles at base = 3° 30' . . . . 8.78648G 

To tan. | their difference = 1° 56J' . . . 8.53053G 




Adding half the difference to half the sum, the larger angle, 
A, is found to be 5° 26 J ; ; subtracting half the difference from 
half the sum, the smaller angle, D, is found to be 1° 33-J 7 . The 

93 



94 &0W TO PROCEED WHEN THE P. C. IS INACCESSIBLE. 



length of the side A D may be found in like manner by trigo- 
nometrical proportion; or, perhaps more simply, thus: — 

BD X nat. cos. D = DF = 279.2. 
B A X nat. cos. A = AF = 79.6. 
AF + FD = AD = 358.8. 

We are now prepared, from our point A, to deflect the angle 
5° 26£ r K, and lay out the line A D to the point D on the curve. 
Moving the instrument to that point, and backsighting to A, a 
deflection of 1° 33| r R places the telescope on line DB; a fur- 
ther deflection of 7° places it in tangent at D, and the curve 
may thence be traced in both directions. 

2. Second Method. — Having, as in the first method, 
judged that stake 27, marked D, must fall on accessible ground, 
and thus determined the central angle subtended by the arc 
B D, refer to Table XVI. for the ap. dist. of a 1° curve, corre- 
sponding to 14°, the given angle. It proves to be 703.5 feet. 
One-fifth of this, 140.7 feet, is the tangent or apex distance, 
BC, of a 5° curve, which may be measured on the ground. 
Moving the instrument to C, turning 14° E, and laying off the 
line C D = B C, the point D on the curve is ascertained. 

3. The preceding methods are manifestly applicable to the 
ends also of curves, as well as the beginnings. A case not 

unfrequent in practice may be added in 
conclusion of the subject. 

Suppose a 2° curve terminating at C, in 
marsh or stream not measurable directly. 
Let C fall at stake 32.20. At any con- 
venient point A, say stake 29, place the 
transit with telescope in tangent. The 
arc AC, = 320 feet, includes an angle of 
6° 24'. The ap. dist. of a 1° curve corre- 
sponding to this angle in Table XYI. is 
320.34 feet; that of a 2° curve is therefore 
160.2 =AB. Move to B, deflect 6° 24/ R 
into the range of the terminal tangent, 
and fix E on the opposite shore. Fix also 
D, and note the angle EBD. Move to 
E. Measure the angle DEB, and the distance D E. The tri- 
angle BED may then be solved. If BE is found to be 070 
feet, C E = 670 — 160.2 = 509.8, and stake E = 32.20 + 509.8, 
*- say 37.30. 





BOW TO PROCEED WEEN THE P. C. C. IS INACCESSIBLE. 95 



XXVII. 

HOW TO PKOCEED WHEN THE P. C. C. IS INAC- 
CESSIBLE. 

1. Suppose a 4° curve, 
A B, compounding at B into 
a 6° curve B C. 

First Method. — Place 
the transit at any point A, 
say stake 34. Let the pro- 
posed P. C. C. fall at stake 
36.25. Assume that we wish 
to reach C, on the second 
curve, by means of the 
straight line ADC. The 
arc AB, covering 225 feet 
of a 4° curve, subtends an 
angle of 9°. A D is half the chord of twice this angle. 

By Table XVI., the chord of 18° on a 1° curve is 1,792.7 feet. 
That of a 4° curve is therefore 448.2 feet, half of which = 
224.1, =AD. The mid. ord^of 18° on a 1° curve, by the same 
table, is 70.54 feet; one-fourth of which, or 17.635, is the mid. 
ord. B D, corresponding to the same angle on a 4° curve. In 
order to find what angle on the 6° curve this mid. ord. B D, = 
17.635 feet, corresponds to, multiply it by 6, and seek the prod- 
uct, 105.81, in Table XVI., where it is found, nearly enough 
for field-practice, opposite the angle 22° 04'. The chord of that 
angle, on a 1° curve, is seen at the same time in the adjoining 
column to be 2,193.2 feet; on a 6° curve it is therefore 365.5 
feet, one-half of which, = 182.75 feet, = DC, and one-half of 
22° 04 / = 11° 02', = the angle covered by the arc B C. Thus 
are found the angle at A = 9°, the angle at C = 11° 02', and 
the distance AC = 224.1 + 182.75, = 406.85 feet. The angle 
11° 02' corresponds to a length of 1.84 feet on the 6° curve; 
C, therefore, falls at stake 36.25 + 1.84 = 38.09. With these 
data the field-work is obvious. 

2. Second Method. — Having reached the point A, and 
determined the arc A B = 9°, as above, find in Table XVI. the 
ap. dist. 450.95 feet, corresponding to the given arc, one-fourth 



96 TO SHIFT A P. C. 

of which = 112.7 feet, = ap. disk for the 4° curve. Move to 
E, deflect 9° R ; range out the line E F, made up of E B = A E 
= 112.7 feet, and 13 F any convenient distance, say 90 feet. 
This 90 feet is the assumed ap. dist. of some unknown angle on 
the 6° curve. To find the angle, multiply 90 hy 6, and seek the 
product, 540, in the AD column of Table XVL, where it is 
found opposite 10° 46'. By moving then to F, deflecting 10° 
46' K, and measuring F C = 90 feet, the point C is fixed on the 
second curve. 

3. Should unexpected obstacles be met in carrying out either 
of these plans, the triangles A G C or E G F may be solved, 
and the point C fixed by means of the lines AG, G C. 

4. The application of the foregoing methods to turning 
obstacles on simple curves needs no special instance. 



XXVIII. 




TO SHIFT A P. C. SO THAT THE CURVE SHALL 
TERMINATE IN A GIVEN TANGENT. 

1. Suppose a 3° curve AB to 
have been located, containing an 
angle of 44° 26', and ending in 
tangent B E : required, that it 
shall end in tangent D F, parallel 
to B E. It is plain, from the 
diagram, that if the curve and 
its initial tangent be moved forward, like the blade of a ska f e, 
until the terminal tangent merges in D F, the P. T. will hr,ve 
traversed the line BD, equal and parallel to AC. If, there- 
fore, on the ground at B, the angle E B D, equal to the whole 
angle consumed by the curve, in this case 44° 26', be laid off 
to the right, and the distance B D to the range of the proposed 
terminal tangent be measured, the equal distance A C, from 
the 'original to the required P. C, is thus directly ascertained. 
Should such direct measurement be impracticable, range out 
the tangent BE, and, at any convenient point, measure the 
distance from it square across to the proposed terminal tan- 
gent D F, say 56 feet. Then in the right triangle BED, mak- 
ing B D radius, we have given the angle at B = 44° 26', and 



TO SUBSTITUTE A CURVE OF DIFFERENT RADIUS. 97 

the sine E D = 56 feet. Hence, by trigonometry, ED-f nat. 
sin. 44° 26', or 56 4- 0.7, = B D = 80 feet, = distance A C along 
the initial tangent, from the erroneous to the correct P. C. 

2. This problem occurs more frequently than any other in 
the field; and the young engineer should have it by heart, that 
the distance square across between terminal tangents, divided 
by the natural sine of the total angle turned, will give him the 
distance he is to advance or recede with his P. C. to make a fit. 

3. Excepting on precarious rocky steeps, city streets, or like 
exact confines, to strike within two feet of any point desig- 
nated in the project, may be considered striking the mark. 
Astronomical nicety, whether with transit or level, in an ordi- 
nary railroad location, is mere waste of time. 

4. The observant reader will not fail to perceive that the 
foregoing rule applies to systems of curves, or to compound 
lines also, the angle EBD being the angle included between 
the initial and terminal tangents, let what flexures or indirec- 
tions soever have been interposed ; and that, if the angle re- 
ferred to be either 180° or 360°, adjustment by shift of P. C. is 
impracticable. In those cases, a change of radius becomes 
necessary. 



XXIX. 

TO SUBSTITUTE FOR A CURVE ALREADY LOCATED, 
ONE OF DIFFERENT RADIUS, BEGINNING AT THE 
SAME POINT, CONTAINING THE SAME ANGLE, AND 
ENDING IN A FIXED TERMINAL TANGENT. 

1. Suppose the 4° curve AB, 
containing an angle of 32° 20', to 
have been located, and that it is 
required to substitute for it an- 
other curve A C, which shall end 
in a parallel tangent C F, 60 feet 
to the right. 

First Method. — Find the 
length of the long chord AC,= 

A B + B C. Referring to Table XVI. , the chord of a 1° curve 
for 32° 20' is seen to be 3,190.8 feet; that of a 4° curve, there- 




98 TO FIND THE POINT AT WHICH TO COMPOUND. 

fore, = 797.7 feet, say 798 feet, = AB. To find BC, solve 
the triangle B D C, observing that the angle DBC = B AI = 
one-half of the central angle 32° 20', = 16° 10', and that D C 
= 60 feet. Then D C -r- nat. sin. 16° 10 / = 60 -f- .278 = say 
216 feet, = BC. Hence AC = AB + B C = 798 + 216 = 
1,014 feet. 

Having thus found the length of chord A C, the radius and 
rate of curvature may be deduced as in X. 

Or, dividing the tabular chord of 32° 20' by chord AC = 
1,014, the degree of the required curve is ascertained directly 
to be 3.15, equivalent to 3° 09'. 

2. Second Method. — Find the apex distance AH,=AI 
+ I H. The tabular ap. dist. of 32° 20' divided by 4 gives A 1 
= 415 feet. In the triangle K D C, the side DC -f nat. sin. 
K = 60 -^ nat. sin. 32° 20', = 112 feet = KC = IH. Then 
AH == A I + IH = 415 + 112 = 527 feet; and the tabular 
ap. dist. 1,661 -f- 527 gives 3.15, equivalent to 3° 09', the degree 
of the required curve A C, as before. 



XXX. 



HAVING LOCATED A CURVE A B C, TO FIND THE 
POINT B AT WHICH TO COMPOUND INTO ANOTHER 
CURVE OF GIVEN RADIUS, WHICH SHALL END IN 
TANGENT E F, PARALLEL TO THE TERMINAL 
TANGENT OF THE ORIGINAL CURVE, AND A GIVEN 
DISTANCE FROM IT. 

1. To find B, the angle BIC must be 
found. Call the given distance between 
tangents D; the larger radius, R; the 
smaller one, r; the required angle, a. 
Then, referring to the figure, observe 
that in the triangle I H K, I H being ra- 
dius, IK is the cosine a; i.e., IK -f- IH 
= nat. cosine a. But I H = R — r; IK 
= IC-KC, and KC = KF or HE 

+ F C, = r + D ; i.e., I K = R — r — D. Hence nat. cosine 

a = (R - r - D) -5- (K .- r) = 1 - D -s- (R - r). 




TO SHIFT A P. C. C. 99 

The same reasoning would apply if A B E were the curve 
first located, and a terminal curve of larger radius required to 
be put in. 

2. We have, then, the following general rule for such cases: 
Divide the perpendicular distance between terminal tangents 
by the difference of the radii, and subtract the quotient from 
unity; the remainder is the natural cosine of the angle of re- 
treat along the located curve to the required P. C. C. 

Example. 

3. A 3° curve on the ground, to find the P. C. C. of a 5° curve 
striking 27 feet to the right. Here D — 27; R - r= 1,910 - 
1,146, =764; D-^ (R - r) = 27-^764, =.03534; and 1 - 
.03534=. 96466 = nat. cosine 15° 17'. We must go back 
therefore, 509 feet on the 3° curve, to compound into the 5° 
curve. Had the 5° curve been located first, we must have 
gone back 306 feet to begin the 3 ° curve which should strike 27 
feet to the left. In either case, time might be saved by moving 
directly from E to C, or the reverse, and spotting in the curve 
backwards. To do this, we have in the right triangle F E C, 
the angle E = half of 15° 17', = 7° 38J', and the side F C = 27 
feet. Then E C = 27 +nat. sin. 7° 38 J', = 203 feet; and if E 
were stake 54.20 on the 5° curve, B would fall at stake 54.20 
-3.06, = 51.14; and C, the P. T. of the 3° curve, at 51.14 + 
5,09, = stake 56.23. 



XXXI. 

TO SHIFT A P. C. C, SO THAT THE TERMINAL 
BRANCH OF THE CURVE SHALL END IN A 
GIVEN TANGENT. 

First Case: the terminal branch a 

having the shorter radius. /N^ 

1. Suppose the compound curve / ^s. 

A C N located, and that it is required / \>p 

to fix a new P. C. C. at B, from which / 

the terminal branch B M shall merge yf x / 
in tangent M L, a given distance from / j Jf~ 
NO. To fix B, the central angle //! 

B H M of the new terminal branch jt"f — i< ~ 

must be found, and substituted for 

C I N. Call the longer radius R; the shorter one, r; the dis- 



100 



TO SHIFT A p. a a 



tance asunder of the terminal tangents, D ; the central angle, 
CIN, =IEK, of the located terminal branch, b; and the 
central angle, B H M, = HE F, to be substituted for it, a. 

In the right triangle, E I K, E K = E I cos. I E K = (R — r) 
cos. b. 

In the right triangle H F E, E F = E II cos. II E F = (R — r) 
cos. a. 

Also, FK = LO = D, since each is equal to r — K L. 

ThenEF = EK — FK; i.e., (R — r) cos. a==(R — r) cos. b 
-D. 

Hence nat. cosine a = nat. cosine b — [D -f- (R — r) ]. 

Were the curve B M located, and the curve C N" to be substi- 
tuted for it, — that is to say, were a given and b required, — 
we should have, by transposition, nat. cos. b = nat. cos. a -f 
D -f- (R - r). 

Example. 

A 3°, compounding into a 5° curve at C, which consumes an 
anyle CIN, =30° 22', and ends in a tangent, NO, which is 
found, by measurement of L O, to be 34 feet too far to the left. 

Here, D = 34, R = 1,910, r = 1,146, b = 30° 22 r ; and, by 
the solution, nat. cos. a = nat. cos. 30° 22' — 34 -^ [1,910 -- 
1,146] = 0.8628 — (34 -f- 764). 



34 

764 

.0445 



log. 1.531479 
log. 2.883093 



lo- 2.648380 



Then 0.8628 — 0.0445 = 0.8183 = cos. 35° 05', = angle a , 
a — 6 = BUM — CIN=BEC 
= the angle of retreat from the 
erroneous V. C. C. = 35° 05 r — 
30° 22' = 4° 43', equivalent to VJ 
feet, on the 3° curve, from C to B. 
2. Second Case: the terminal 
branch having the longer radius. 

Let BN represent the terminal 

branch located with central angle 

I K O = 6, and suppose it required 

to determine the new arc CM, 

with central angle IEF=a. Call the longer radius R, the 

shorter one r; the distance L N" between tangents, D. In the 




TO SHIFT A p. c. a 101 

right triangle IKO, KO=KI cos. IKO = (R— - r) cos. b. 
In the right triangle FIE, EF = EI cos. IEF = (It — rj 
cos. a. Also, EH = LN=D, since each is equal toR-KL. 

Then EF = EH + HF = EH + KO; i.e., (R — r) cos. 
a = (R — r) cos. b -\-D. Hence nat. cos. a = nat. cos. b -f- 
D -*- (R - r). 

Were the curve C M located, and the curve BN to be sub- 
stituted for it, that is to say, were a given and b required, 
we should have, by transposition, nat. cos. b = nat. cos. a — 

D -s- (R - r). 

■ Example. 

A 5° compounding into a 3° curve at B, which consume? 
an angle of 44° 20', and terminates at X, 28 feet too far to 
the left. Here D = 28, R = 1,910, r = 1,146, b = 44° 20; 
and, by the solution, nat. cos a = nat. cos. 44° 20' -f- 28 
-f- 764. The nat cos. 44 ' 20' = 0.7 ; 28 -f- 764 = log. 

1.447158 — log. 2.883093 — log. 2.5G4C05, corresponding to the 
decimal 0.03665, which, being added to nat. cos. 44° 20', gives 
0.75194, the nat. cos. 41° lo. Then 13 K X — C E M = 44° 20' 
— 41 c 15' = 3° 05'= angle BIC, equivalent on a 5° curve to 
62 feet, which therefore is the distance around the arc from 
B, the erroneous P.C.C., to C, the correct one. 

3. From these formulas the following general rule may be 
drawn: Divide the distance between terminal tangents by the 
difference of the radii, and call the quotient Q. Find the nat- 
ural cosine of the terminal arc already located, and call it C. 
The sum or the difference of Q and C will be the natural 
cosine of the terminal arc to be substituted for that already 
located. With radii in the order R, r, should the terminal 

tangent located strike \ . , r the proposed tangent; or, 

with radii in the order r, R, should the terminal tangent 

located strike < . ., > the proposed tangent, — take the 

! ,. ff | of Q and C for the required cosine. 



102 TO FIND TlIE POINT AT WHICH TO BEGIN A CURVE. 



XXXII. 



HAVING LOCATED A TANGENT, A B, INTERSECTING 
A CURVE, C D, FROM THE CONCAVE SIDE, TO 
FIND THE POINT E ON SAID CURVE AT WHICH 
TO BEGIN A CURVE OF GIVEN RADIUS WHICH 
SHALL MERGE IN THE LOCATED TANGENT. 

/ 1. Place the transit at the 

intersection point B. Set 
points at equal distances 
therefrom in both directions 
on the curve already located, 
by means of which the direc- 
tion of a tangent to that curve 
at B may be fixed, and the 
angle FBA measured. Call 
that angle a; and, as shown 
in the figure, suppose I he lo- 
cated curve to be prolonged in- 
to a terminal tangent, parallel 
with the newly located tan- 
gent A B. Complete the dia- 
gram. Call the larger radius R; the proposed radius, r; the 
central angle of the proposed curve, x. Then, obviously, the 
line AG = R cos. a. It is also equal to (R — r) cos. x -f- r. 
That is to say, R cos. a = (R — r) cos. x -f- r. Hence cos. x 
= (R cos. a — r) -s- (R — r) ; and x — a = angle B GE, sub- 
tended by the arc BE, from which the length of the arc may 
be deduced, and the point E ascertained. 

Example, 
DC, a 1° curve; angle a = 64° 32': to connect with a 
curve. Here cos. x =■ (5,730 X 0.43 - 1,433) ■+■ (5,730 - 1,433) 
= 0.24 == cos. 76° OtV; and x — a = 11° 34 ; , equivalent to a dis- 
tance from B around the 1° curve of 1,157 feet to E, the point 
at which to begin the 4° curve. 




TO LOCATE A Y. 



XXXIII. 

HAVING LOCATED A TANGENT, A B, INTERSECTING 
A CURVE, C IJ, FROM THE CONVEX SIDE, TO FIND 
THE POINT E ON SAID CURVE AT AVHICH TO 
BEGIN A CURVE OF GIVEN RADIUS WHICH 
SHALL MERGE IN THE LOCATED TANGENT. 



! -^ d 



1. This problem is analo- 
gous to the preceding one. 
The preparatory steps are the 
same in both. Having found 
the angle «, however, it will 
be manifest to the attentive 
reader, that, in this case, R 
cos. a = (li + r) cos. x -f- r. 
Hence cos. x = (R cos. a — r) 
-MR + r). 

Example. 

2. D C, a 1° curve; angle a = 64° 32 / : to connect with a 4° 
curve. Here cos. x = (5,730 X 0.43 - 1,433) *- (5,730 + 1,433) 
= 0.1439 = cos. 81° 43'; and x — a = 17° 11', equivalent to a 
distance from B around the l°~curve of 1,718 feet to E, the 
point at which to start the 4° curve. 




XXXIV. 

TO LOCATE A Y. 

1. The processes of the 
two former problems may 
be adopted. In this case 
the angle a vanishes, and 
the cos. x clearly is equal to 
(R-r)-HK + r). 

2. Another solution of the 
Y problem is as follows: — 

Draw the tangent E D in- 
tersecting the tangent B A. Then is B D = D A, for the rea- 



B 




D 




A 




/ 
/ 

i 




\ 


/ 

\ 1 

V \ / 


\ 
\ 

\ 


c 


i 

1 
1 
1 








^V 




1 
\ 
\ 








V 




i 















104 



TO LOCATE A Y. 



son that each is equal to D E. Make GF = R + r, the diame- 
ter of a semicircle. Said semicircle touches tangent B A at D, 
its middle point; and D E being perpendicular to G F, we have 
by geometry GE : DE :: DE : EF; i.e., GE X EF, or R X 
r, = D E-. Hence DE = BD=^DA = |/RXr = R tan. i x, 
and we are thus enabled to fix the points E and A. 

8. In the two foregoing problems, the angle consumed by 
curve E A is = 180° — x. 

Example. 

BE, a 2i° curve located; BA, a tangent: to complete the 
Y with a 6° curve, E A. 

By the first method, cos. x = (R — r) -f- (R + r) = (2,292 
— 955) -f- (2,292 + 955) == 1,337 -^ 3,247 = log. 3.126131 — 
lo(j. 3.511482 = 1.014649, which corresponds to log. cos. 
9.614649, or to the decimal number 0.4118, indicating in either 
case the angle 65° 41' = x. 

D E = B I) = I) A = R tan. $x = 2,292 X 0.6455 = 1,479.4. 
DE may be found also by reference to Table XVI., where the 
ap. dist. of a 1° curve for 65° 41 ; is seen to be 3,698.6. Dividing 
this number by 2J, we have 1,479.4, as above. 

Or, by the second method, — 

D E = VR~X~r = V218S860 = 1,479.4. 

Having thus the means 
of fixing points E, D, and 
A, the curve E A can be 
laid down. 

4. 7/BA is curved con- 
vex to the Y, construct 
the figure as in margin, 
and reason thus: — 

In the triangle EGF, 
formed by lines connect- 
ing the curve-centres, the 
sides are respectively 
equal to the sums of the 
contiguous radii: the 
angles may therefore be 
found as in Case III., 
Trigonometry. 
Lines drawn bisecting the central angles of the several 




TO LOCATE A Y. 105 

curves will pass through the points of intersection of the tan- 
gents to those curves severally. But lines so drawn in this 
case hisect also the angles of a triangle, and, demonstrably 
by geometry, meet in one point equidistant from the three 
sides of the triangle. That point, therefore, must be a com- 
mon P. I. for all the curves, and that equidistance the "ap. 
dist." length common to them all. 

Example. 
Given B A, a 3°, and B C, a 4° curve: to complete the Y with 
a 5° curve, C A. 

EF = 1,910 + 1,140 = 3,056. 
GF = 1,433 + 1,140 = 2,579. 
EG = 1,910 + 1,433 = 3,343. 

Then, by Case III., Trigonometry, — 

As EG, 3,343 log. (a. c.) 6.475804 

Is to E F + G F, 5,635 .... log. . . 3.750S94 
SoisEF— GF, 477 . . . . log. . . 2.678518 

To diff. of segments of E G, 804 2.905276 

Adding half the difference to half the sum of the segments 
of the base EG, we shall have the greater of them; i.e., 
(3,343 -f 804) + 2 --= 2,073.5, which is the cos. E, E F being 
radius. Hence 2,073.5 H- 3,056 = log. 3.316704 — log. 
3.485153 = 9.831551 = cos. 47° 16' = E. By Table XVI., the 
ap. dist. of a 1° curve corresponding to this angle is 2,507 3: 
that of a 3° curve, therefore, is 835.8 — the common ap dist. 
B D or D A. Multiplying the common ap. dist. by 4, we shall 
find opposite the product in Table XYL the central angle of 
the 4° curve to be 60° 32 r ; multiplying it by 5, we find, in like 
manner, the central angle of the 5° curve to be 72° 12'. Arc 
B A, = 47° 16', is equivalent to 1,575 feet on the 3° curve; arc 
B C, = 60° 32', is equivalent to 1,513 feet on the 4° curve. 
Points being thus fixed at A and C, curve C A can be laid on 
the ground. 

5. If curve B A is concave to the Y, the radii being given, 
construct the figure as follows : — 

First draw the triangle GFE, the sides of which are obvi- 
ously derived from the given radii. Prolong the sides E G and 
E F indefinitely. Bisect the exterior angles at G and F with 



106 



TO LOCATE A Y. 



lines meeting at D, and from D let fall perpendiculars on EB, 
EA, and GF. Then, comparing triangles GBD, GCD, the 
angles at G are equal by construction; the angles at B and C 
are right angles, the side GD common. Hence the triangles 
are equal in all their parts: B G = G C, and BD = DC. By 
like reasoning, it appears that CF = FA, and DA = DC. 
The point D being equidistant from the right lines E B, E A, 
which limit angle E, a line bisecting that angle will strike 
point D. 




6. It may be remarked, therefore, that lines bisecting the 
vertical angle and the exterior angles contained between the 
base and the prolongation of the sides of any triangle, will 
meet in a point equidistant from the base and the said prolon- 
gations. We thus have in the figure all the conditions for fit- 
ness of the curves. It remains only to solve the triangle G F E, 
seeing that from its angles the required central angles can be 
obtained. 



Example. 






B A, a 1°, B C, a 6° curve, located : to complete the Y with 
an 8° curve, C A. 



TO LOCATE A TANGENT TO A CURVE. 107 

In triangle GFE,- 

EF = 5,730 — 717 = 5,013. 
EG = 5,730 — 955 = 4,775. 
GF= 955 + 717 = 1,672. 

Then, by Case III., Trigonometry, — 

AsEF . . . 5,013. . . . log. (a. c.) 6.299902 
Is to E G + G F, 6,447 .... log. . . 3.809358 
SoisEG — GF, 3,103 . . . . log. . . 3.491782 

To diff. seg. of base, 3,991 . . . log. . . 3.601042 

The longer segment, therefore, is 4,502; the shorter, 511. 
Cos. E = the longer segment divided by E G = 4,502 4- 4,775 = 
log. 3.653405 — 3.678973 = 9 974432 = cos. 19° 28' = angle E. 

Cos. GFE = the shorter segment divided by GF = 511 -~ 
1,672 = log. 2.708421 — log. 3.223236 = 9.485185 = cos. 72° 
W = angle GFE. 

The central angle, B G C, of the 6° curve, is equal to 180 — 
F G E = the sum of the angles at E and F = 72° 12' + 19° 
28' = 91° 40', making the arc B C = 1,528 feet. The arc B A, 
equivalent to 19° 28' of a 1° curve, = 1,947 feet. Points C and 
A being thus ascertained, curve AC maybe located. It will 
consume an angle = 180° — 72° 12' = 107° 48', equivalent, on 
an 8° curve, to 1,347.5 feet. 



XXXV. 

fO LOCATE A TANGENT TO A CURVE FROM Atf 
OUTSIDE FIXED POINT. 

1. If the ground is open, and the curve can be seen from the 
Sxed point, it may be marked by stakes or poles at short inter- 
vals, and the tangent laid off without more ado. 

2. Suppose, however, that on cumbered ground a trial tan- 
gent, A B, has been run out, intersecting the curve at B : it is 
required then to find the angle B A E, in order that the true 
tangent A E may be laid down. 



108 TO SUBSTITUTE A CURVE. 

Example. 

AB = 1,500 feet ; D II B, a 4° curve ; angle F B D = 20° 13'. 

First, the angle FBD, between a tangent and a chord, is 
equal to half the central angle subtended by the same chord. 
Angle D C B, therefore, = 40° 26'. By Table XYI. , the chord 
of 40° 26', for a 1° curve, = 3,960.2 feet; for a 4° curve, it is, 
say, 990 feet = DB; and D I = I B = 495 feet. The mid. ord. 
H I is, in like manner, found to be 88.25 feet. Deducting this 
from the radius of the 4° curve, we have I C = 1,344.4 feet. 

Then IC-7-IA = £cm. I AC; i.e., 1,344.4 -J- (495 + l,500j 
= 0.674 = tan. 33° 59' = angle I A C. 



Next, by geometry, the proposed tangent AE= \/A D X A B 
= a/2,490 X 1,500 = 1,932.6; and E CvAE = ta. EAC = 
l,432.69-i-l,932.6 = 0.7413 = tan. 36°33 / = angleE AC. Then 
EAC — IAC = 36° 33' — 33° 59' = 2° 34' = angle B A E, the 
angle required, which can accordingly be laid off from the fixed 
point A, and the tangent located. 



XXXVI. 

TO SUBSTITUTE A CURVE OF GIVEN RADIUS FOR 
A TANGENT CONNECTING TWO CURVES. 

Example. 

1. A B, a 4° curve; BC = 774 feet; CD, a 6° curve: to put 
in the 1° curve, E F. 

Sketch the figure as in margin, HK being parallel and equal 
to BC. Then KG = BG — BK or CH = 1,433 — 955 = 
478 feet; KH -r- GK = 774 -f- 478 = 1.62 = tan. 58° 19' = 
angle KGH; and K II -f- sin. 58° 19' =774-7-0.851 = 909.6 
feet = G H 



TO RUN A TANGENT TO TWO CURVES. 



109 




In the triangle GHI we have then the sides given; namely 
G H = say, 910 feet, II I = 5,730 — 955 = 4,775 feet, and G I 
= 5,730 — 1,433 = 4,297 feet : to 
find the angles. 

Under Case 3, Trigonometry 

(in.), in : IG + Gil :: IG 

-GH : IL-LH; i.e., 4,775 : 
«5,207 : : 3,387 : 3,693, the differ- 
ence of the segments into which 
ihe hase 1 II is divided hy a 
perpendicular from G. Adding 
half the difference of the seg- 
ments thus found to half their sum, the longer segment, I L, is 
found to be 4,234 feet ; subtracting half the difference from 
half the sum, the shorter segment, L II, is found to be 541 feet. 
Then H L -f- II G = 541 -f- 910 = 0.5945 = cos. 53° 31< = 
angle GUI. In like manner, dividing IL by IG, we find the 
angle GUI to be 9° 49'. The sum of these angles = angle 
EG II = 63° 20', for the reason that each is equal to ISO — 
II G I. Finally, E G II — K G II = 03° 20' — 5S° 19' = 5° 01' 
= angle EGB, equivalent to a distance from B of 125 feet 
around the 4° curve to the P. C. C. at E; and G I II — EGB 
= 9° 49' — 5° 01' = 4° 4S' = angle CIIF, equivalent to a 
distance from C of 80 feet around the 6° curve to the P. C. C. 
atF. 



XXXVII. 

TO RUN A TANGENT TO TWO CURVES ALREADY 
LOCATED. 




then be 
the line 



1. If one curve be visible 
from the other, or if both 
be visible from some inter- 
mediate point, mark them 
on the ground with stakes 
at short intervals. The 
points M or L in the range 
of the required tangent may 

fixed by one or two trial settings ©f the transit, and 

put in. 



HO TO RUN A TANGENT TO TWO CURVES. 

2. Should obstacles prohibit this plan, measure any con- 
venient line, FG or B CD, from one to the other curve, and, 
completing the traverse AFGE or ABODE, determine 
thence the bearings and distances asunder of the centres A 
and E. The right triangle A E K, in which E K = the sum of 
the radii, may then be solved, and the points II and I ascer- 
tained as in the following example : — 

Example. 

F B, a 4° curve ; G D, a 6° curve. N. S. E. Wo 

A B, N. 20° E., 1,433 feet . • 1,346.6 490.0 

B C, East, 3,570 feet . . - 3,570.0 

C D, N. 34° E., 1,800 feet . . 1,492.2 1,006.2 

DE, N. 45° W., 955 feet . . 675.2 - 675.2 

3,514.0 5,066.2 675.2 

Total northing, 3,514 feet; total easting, 4,391 feet. 

Then 4,391 -f- 3,514 = 1.2496 = tan. 51° 20' = bearing 
AE; and 4,391 -f- sin. 51° 20' = 5,624 feet = distance A E. 
Also, EK -^ AE = (1,433 + 955) -f- 5,624 = sin. 25° 08' = 
angle EAK; and angle AEK = 90° 00' — 25° 08'= 64° 52'. 
Hence the bearing of A K or HI is N. 76°28'E., and that of 
AH or IE, N. 13°32'W. 

Since AB bears N. 20° E., the angle II AB = 20° 00' + 13 
32'== 33° 32', equivalent to a distance of 838 feet from B arou 
the 4° curve to the required P. T. at II ; and, since DE be 
N. 45° 00' W., the angle I E D = 45° 00' — 13° 32/ = 31° 28 
equivalent to a distance of 524 feet from D around the 6° curve 
to the required P. C. at I. 

3. Should the curves turn in the same direction, the side 
EK of the triangle AEK is equal to the difference of the 
radii instead of their sum. In other respects, the method 
exemplified will apply to that case also. 

4. The preceding solution may be useful as an exercise, 
But the problem is one of rare occurrence, and the condition 
must be extraordinary which prevent a close approximation 
at least, to the true line in the field. The better way in actual 
practice, then, is to run out a trial tangent as nearly right as 
V*ossible. If it errs by passing outside the objective curve 
close with a compound (XXIX.); if that error be inadmissible, 
or if it errs by cutting the objective curve, measure the miss, 
,and divide it by the length of the jtriaj tangent. The quotient 



13 ; 

uiiii 
pai> 

Oft 



TO RUN A TANGENT TO TWO CURVES. HI 

will be the natural tangent of the angle of retreat or advance 
on the first curve required to make the tangent fit. 

5. A still closer adjustment would be, after determining the 
angle approximately as above, to find the "tangents" corre- 
sponding to it for the two curves in Table XVI. Subtract the 
sum of these tangents from the length of the trial line, if it 
cuts the objective curve; add the sum, if it passes out&ide. 
With the number thus found, divide the measured amount of 
error for the tangent of the angle of retreat or advance, as the 
case may be. 

G. Suppose, for illustration, that a trial tangent, bearing by 
needle N. 54° 30' E., is run out from stake 24.80 of a 4° curve, 
intending to touch a G°, but is found to cut it. Suppose fur- 
ther that the objective G° curve was laid down and numbered 
i:i the direction of approach towards the 4° curve; that its P. 
C. is stake 25.10, and the magnetic bearing of its initial tan- 
gent S. 30° 30' W. The angle, then, between the bearing of the 
trial tangent and that of the initial tangent of the G° curve, is 
24°, corresponding to a distance of 400 feet on the latter curve. 
At stake 25.10 + 4.0 == 29.10, therefore, a tangent to the 6° 
curve would be parallel to the trial tangent. Go forward on 
the trial tangent, accordingly, to a point opposite 29.10, and 
measure the distance square across to that plus on the G° 
curve. Assuming the trial tangent to be 2,500 feet long, and 
the amount of the miss to be $7 feet, the nat. tan. of the 
angle of error is 0.0348 = tan. 2°. By the method in (4), this 
calls for a shift of the P. T. 50 feet ahead on the 4° curve, 
making the new P. T. 24.80 + 0.50 = stake 25.30, and ad- 
vances the P. T. of the 6° curve to stake 29.43 of that numera- 
tion. 

The method in (5), applied to this case, brings the angle of 
error 2° 02', instead of 2°, equivalent to a deviation of 1J feet 
scant in half a mile from the line corrected by the method in 
(4), and agreeing exactly with the correction determined by 
the method in (2). 



TRACK PROBLEMS, 
XXXVIII.— LI. 

(Standard Gauge 4 Feet 8J Inches.) 



TRACK PROBLEMS. 



XXXVIII. 

REVERSED CURVES. 
The following problems will be useful in laying off turnouts, 
the adjustment of tracks near stations or shops, and the like; 
but reversed curves should never be used on the main line 
between stations, where they are both objectionable and unne- 
cessary. Ground which allows any permissible location 9t alt 
will allow straight reaches of at least two hundred or three 
hundred feet between curves of contrary flexure; and in every 
case it is worth the small additional outlay to make such a 
location. 



XXXIX. 



TO CONNECT TWO PARALLEL TANGENTS BY A 
REVERSED CURVE HAVING EQUAL RADII. 

1. The radius R, and the perpendicular distance D, between 
the tangents given. ^c 

/ 




JUS 



116 TO CONNECT TWO PARALLEL TANGENTS. 

Draw the tangents, radii, and curves, fixing the P. R. C. . 
midway of D. 

Draw the chords G I, IE, the line B F perpendicular to G I, 
and the line E H in prolongation of radius C E to an intersec- 
tion with B H passed through centre B parallel to tangents. 

That I falls midway of D, follows from the necessary sym- 
metry of the figure ; and G I E must be a straight line, because 
the radii B I, I C, perpendicular to a common tangent at the 
same point, form a straight line, to which the chords G I, IE, 
are equally inclined. 



CH^CB = cos. A; but C H = 2 R — D, and C B = 2 R, 

.'. cos. A = (2R-D)-f-2R. 

B H = B C sin. A = 2 R sin. A; 

GF = Rsm. i A; GE = 4GF. 

.-. GE = 4 R sin. i A, and GI or IE = 2 R sin. i A. 



Observe, that, in the right triangles G K E and B G F, the 
angles at G and B are each equal to i A: hence the triangles 
are similar. 

Example. 

R = 800 feet, D = 24 feet. 
To find angle A. 



R. 

the 
,ngles 



Cos. A = (2 R — D) -f- 2 R = 1,576 -f- 1,600 = 0.985 = nat. 
cos. 9° m f . 

BH may then be found = 2 R sin. A = 1,600 X 0.1725 = 
276 feet, and laid off from the P. C. at G to K, the point E 
being fixed by a right angle from K. 

Or GE may be found = 4 R sin. i A = 3,200 X 0.866 = 
277.1 feet, and laid off from G to E, the point I being fixed 
138.5 feet from G, and angle KGE made equal to half of A = 
4° 58'. 

2. The distances G K and D given, to find R. 

In triangle GKE, KE == D. 

D -T- GK = tan. }A;D-f- sin. }A = GE; and GE -f- 
sin. i A = 4 R. 

Or, having found G E, we have from the congruity of trian- 
gles GKE, BFG, 

D : GE :: |GEorGF : R. 
.'. R = GE 2 -i-4D. 



TO CONNECT TWO PARALLEL TANGENTS. 



117 



Example, 
G K = 300 feet, D = 28 feet. 

D-i-GK Log. 28 ... . 1.447158 

Log. 300 ... . 2.477121 

= Tan. i A . 5° 20 / . . . . 8.970037 

D -f- sin. i A . . . . Log. 28 ... . 1.447158 

Sin. 5° 20' . . . . 8.968249 

= GE . . 301.24 . . . .2.478909 

GE-fskH . . .Sin. 5° 20' . . . .8.968249 

= 4 R . . 3,241 .... 3.510660 
.\ R = 810.2. 



XL. 

TO CONNECT TWO PARALLEL TANGENTS BY A 
REVERSED CURVE HAVING UNEQUAL RADII. 




1. Given the perpendicular distance, D, between two paraL 
(el tangents, and the unequal radii, R and r, of a reversed 
curve, to find the central angles, A, the chords, and the 
straight reach, G K, of the curve. 



118 TO CONNECT TWO PARALLEL TANGENTS. 

Cos. A = CH-i-BC; but CH = (K + r)-D, and 

BC = R + r. 

.\ Cos. A = (R + r — D) -i- (R + r). 

The straight reach GK = BH=(R-f-r) sin. A. 
The sum of the chords G E = G K -i- cos. i A. 

GI = 2Ksm.{A. 
IE = 2 r sin. i A = GE — GI. 

Example. 
D = 28, R = 955, r = 574. 

Cos. A = (R + r — D) -f- (R + r) = 1,501 -f- 1,52&> 

1,501 . . . log. 3.176381 
1,529 . . . log. 3.184407 

Cos. A, 10° 59' . , . . . 9.991974 

GK= (R + r) sin. A. 

R + r, 1,529 . . . log. 3.184407 
Sin. A, 10° 59' . . . log. 9.279948 

GK = 291.3 2.464355 

GE = GK-f- cos. i A. 

GK, 291.3 . . . log. 2.464355 
Cos. $ A, 5° 2ty . . . log. 9.998014 

GE = 292.6 2.466341 

GI = 2R sin. i A. 

2 R, 1,910 . . . log. 3.281033 
Sin. i A, 5° 29£' . . . log. 8.980916 

' GI = 182.8 2.261949 

IE = GE — GI = 292.6 — 182.8 = 109.8. 

2. The distances GK and D, and one of the unequal 
radii, R, given, to find the other radius, r, and tbft central 
angles, A. 



REVERSED CURVE WITH UNEQUAL ANGLES. 119 

Example. 
GK = 422, D = 30, R = 2,292. 

Tan. iA = D-r-GK. 

D = 30 . . . log. 1.477121 
GK = 422 . . . log. 2.625312 

Tan. i A, 4° 04' ... . . 8.851809 
.\ A = 8° 08'. 

GE = D-fsk \ A. 

D = 30 . . . log. 1.477121 
Sin. i A, 4° 04' . . . log. 8.850751 

GE = 423 2.626370 

GI = 2Rsm. | A. 

2 R =4,584 . . . log. 3.661245 
Sin. i A, 4° 04' . . . log. 8.850751 

G 1 = 325.1 2.511996 

GE-G 1=423-325=98 = 1 E. 

r = § lE+sin. f A. 

UE = 49 . . . log. 1.690196 
Sin. \ A, 4° 04' . . . log. 8.850751 

r = 691 2.839445 



XLI. 

A REVERSED CURVE HAVING UNEQUAL ANGLES. 

Given the angles A and B, and the length A B of a straight 
line connecting two diverging tangents, to find the radius of 
a reversed curve to close the angles. 

AI=RX&m. \A\ Bl = RXtan. J B. 
/. AB=RX(fcm. iA+tan. § B). 
.'. R=A B-r (tan. \ A+tan. £ B). 



T20 REVERSED CURVE BETWEEN FIXED POINTS. 



Example. 



A = 16°, B = 10°, AB = 840. 




AB, 840 o log. 2.924279 

| A = 8°, nat. tan. 0.14054 

i B = 5°, nat. tan. 0.08749 
Tan. i A + tan. i B = 0.22803 . . . log. 1.357992 

R = 3884 3.566287 

This solution will apply also to the finding of the maximum 
radius for a simple curve which shall connect three tangents. 



XLII. 



■ 



A REVERSED CURVE BETWEEN FIXED POINTS. 

Given the angles N and K, and the length of the straight 
line E F connecting two divergent tangents, to find the radius 
of a reversed curve from E to F, connecting the tangents. 

1. Denote the angle EIC or DIFbyl; the angle C E I, 
complement of N, by n; and the angle DFI, complement of 
K, by k. 

Then, in triangle E C I, — 



EC : CI : : sin. I : sin. n. 



E C X sin. n = C I X sin. I. 



REVERSED CURVE BETWEEN FIXED POINTS. 



121 



Also, in triangle D F I, — 
D F : D I : : sin. I : sin. k. .'.DFX sin. k = T>lXsin.L 

Adding these equations, we have — 

EC X sin. n + DFX sin. k = (CI + DI) X sin. I. 




But E C and D F are each equal to R; sin. n = cos. N tl% 
k = cos. K; andCI + DI = 2R. 
Hence the equation becomes, — 

R X (cos, N + cos. K) = 2EX sin. I. 
,\ sin. I = (cos. N -f- cos. K) -f- 2. 

The foregoing elegant solution is abridged from Hen«ik. 
2. Angle A = 180 — (n -f- I) ; angle B = 180 — ( k + ±,. 
To find radius, draw F H parallel, and E H perpendicular, to 
CD. 
Then E H = E F X sin. I. 

But EH = Efi + GH; EG = R X sin. A; ahdGH = K 
X sin. B. 

.\ EF X sin. I = R X (sin. A + sin. B). 
.-. R = EF X sin. I -f- (sin. A + sin. B). 



122 REVERSED CURVE BETWEEN FIXED POINTS. 

Example. 
E F = 1,400, N = 30°, K = 20°. 

Sin. I = (cos. N + cos. K) -f- 2. 

N = 30°, nat. cos 0.86603 

K = 20°, nat. cos 0.93969 

1.80572 . 
1.80572 -T- 2 = 0.90286 = nat. sin, 64° 32'. 
.-. I = 64° 32'. 

A = 180° — (n + I) = 180° — (60° + 64° 32') = 55° 28'. 
B = 180° — {h + I) = 180° — (70° + 64° 32') = 45° 28 f . 
K = E F X sin. I -r- (sin. A -f sin. B). 

EF = 1,400 log. 3.146128 

Nat. sin. I, 0.90286. . log. T. 955621 

EG = 1,264 3.101749 

A = 55° 28' nat. sin 0.82380 

B = 45° 28' nat. sin 0.71284 

Sin. A + sin. B 1.53664 log. 0.186579 

R = 822.6 2.915170 

3. The young student should bear in mind that the addition 
or subtraction of the logarithms of two natural numbers gives 
a logarithm representing, not the sum or difference, but the 
product or quotient, of such numbers. When, therefore, as in 
the two foregoing cases, the sum or difference of two or more 
trigonometric functions — sines, tangents, and the like — is 
sought, the logarithm of the sum of the natural functions, and 
not the sum of their logarithms, is to be used. If, for example, 
sin. A X sin. B is required, the log. sin. A -|- log. sin. B = the 
logarithm of the product of the sines designated ; but, if sin. A 
+ sin. B is sought, the natural sines of those angles must be 
added together, and the logarithm of the sum of these natural 
functions must be used in making logarithmic calculations. 



TO CONNECT TWO DIVERGENT TANGENTS. 



123 



XLIII. 

TO CONNECT TWO DIVERGENT TANGENTS BY A 
REVERSED CURVE. 




1. ADVANCING TOWARDS THE INTERSECTION OF TANGENTS. 

Given the angle of divergence, N, the initial P. C. at G, 
the distance G H, and the radii R, r, to find the central angles 
A and B. 

GK = CGXcos.N= Rcos. N. 

GL=GHXskN. 

GK-GL=LKorEF,CF being drawn parallel to L E. 

Cos. B = DF-rDC=(r + EF)-(R+r). 
I Angle GCK = 90°-N; angle DC F = 90°-B. 

Angle A = GCK - D C F = (90° - N) - (90°-B) = B 
-N. 

Example. 

N = 24° 30', GH = 854, R = 1,440, r = 1,146. 

GK = R cos. N. 



R = 1,440 
Cos. N, 24° 30' 

GK = 1,310 



log. 3.158362 
log. 9.959023 

. . 3.117385 



124 



TO CONNECT TWO DIVERGENT TANGENTS. 



GL = GH X sin. N. 

Gil = 854 
Sin. N, 24° 30' 



log. 2.031458 
log. 9.017727 



GL = 354 2.549185 

LK or EF = GK — G L = 1,310 — 354 = 956. 
Cos. B = (r + EF) -f- (R + r). 



r + EF = 2,102 . 

R + r = 2,586 . 



Cos. B 3 35° i 



log. 3.322633 
log. 3.412629 

. . 9.910004 



B = 35°38'. 
A = B — N = 



: 350 33/ _ 24° 30' = 11° 08'. 



K\ 



A L 






Sau-J 



T 



V— f 






C 



-Jf 



2. RECEDING FROM THE INTERSECTION OF TANGENTS. 

Given the angle of divergence, N, the initial P. C. at G, 
the distance G H, and the radii R,'r, to find the central angles 
A and B. 

G K = G H X tan. N. 
KC = GC-GK-R-GK. 

LC orEF = KC X cos. N, the line C F being drawn paral- 
lel to L E. 
Cos. B = D F -r- C D = (r + E F) -i- (R + r). 
Angle A manifestly = B -f- N". 



TO SHIFT A P. R. C. 126 

Example. 
N = 18° 30', GH = 920, R = 955, r = 819. 

GK = GHX tan. N. 

G H = 920 . . . log. 2.963788 
Tan. 18° 30' .. . . los. 9.524520 



GK = 307.8 2.488308 

K C = R — G K = 955 — 307.8 = 647.2. 
LCorEF = KC X cos. N. 

KC = 647.2 . . . log. 2.811039 

Cos. N, 18° 30' . . . log. 9.976957 

EF = 613.8 2.787996 



Cos. B = (r + EF)-f(R + r). 

r + EF= 1,432.8 . . . log. 3.156185 

R-fr = 1,774 . . . log. 3.248954 

Cos. B, 36° 08' 9.907231 



B = 36° 08 

A = B + N = 36° 08 / + 18° 30 r = 54° 38'. 



XLIV. 

TO SHIFT A P. R. C. SO THAT THE TERMINAL 
TANGENT SHALL MERGE IN A GIYEN TANGENT 
PARALLEL THERETO. 

Given the reversed curve E F G, ending in tangent GV: to 
find the angle of retreat, A, on the first branch, and the angle 
C of the second branch, ending in tangent HT, parallel to 
GY. 

Measure the error T G = D, perpendicular to the terminal 
tangent. 



126 



TO SHIFT A P. R. C. 



In the figure, draw L K parallel to G V, and passing through 
centre of first branch. 




Then M K = (R + r) X cos. B. 

NL =(R + r) X cos. C. 

WL = GK. 

NL = r + D + GK. 

MK = r + GK. 

NL-MK = D. 

.-. (R + r) X cos. C — (R + r) X cos. B = D. 

.-. (R + r) X cos. C = (R + r) X cos. B + D. 

.-. Cos. C=[(li + r) X cos. B + D)-^(R + r)o 

A = (90° - C) - (90° - B) = B - C. 

Example. 
R = 1,433, r = 819, B = 34° 20 7 , D = 94. 
Cos. C = [(R + r) cos. B + D] -f- (R + r). 



R -f r = 2,252 
B = 34° 20', cos. 



log. 3.352568 
los. 9.910859 



(R + r) cos. B = 1,860 ..... 3.269427 
Add D 94 

■ . log. 3.290925 

. log. 3.352568 



1,954 
(R + r) 



Cos. C, 29° 49' 9.938357 

A = B — C = 34° 20' — 29° 49' = 4° 31'. 



CURVE THROUGH A FIXED POINT. 



127 



XLV. 

TO PASS A CURVE THROUGH A FIXED POINT 
THE ANGLE OF INTERSECTION BEING GIVEN. 




Given the intersection angle, A, of two tangents, to find the 
radius, R, of a curve which shall pass through a point, C; 
the position of said point, with reference to the tangents or the 
point of intersection, being known. 

1. By what data soever point C is located, they may be com- 
muted by simple processes to the form shown in the figure; 
namely, the ordinate B C and the distance I C to apex. Call 
the angle BI C a, and complete the triangle ICO. 



In this triangle, x = ( 18 ° 2 A ) — a = 90° - ( i A + 



a). 



Also, C O : I O : : sin. x : sin. z. 

But C = R; I = — 5^-r. 
cos. J A 



.R: 



R 



cos. i A 



sin. x c* sin. z. 



Hence sin. z ■ 



cos. i A 
solved, and the radius ascertained. 



The triangle ICO may then be 



128 



CURVE THR0UGI1 A FIXED rOINT. 



Example. 

A = 40°, B C = 32 feet, I B = 80 feet. 
Then BC -h IB = 32 -f- 80 = 0.4 nat. tan. 21° 49'; and 
I C = B C -^ nat. sin. 21° 49' = 32 -f- .372 = 86 feet. 
Also, x = 90° — (J A + a) = 90° — 41° 49' = 48° 11'. 



Next, sin. x, 48° 11' 
Divided by cos. \ A, 20° 

= sin. z, 127° 31' 



log. 9.872321 
log. 9.972986 



log. 9.899335 



Or, since the sine of any angle is equal to the sine of its sup- 
plement, the supplement in this case, 52° 29', may be taken 
directly from the logarithmic table, from which supplement 
deducting x, or 48° 11', the remainder is the angle y — 4° 18'. 






Finally, IC = 86 
Multiplied by sin. x 9 48° 11' 

= CD 
And C D divided by sin. y, 4° 18' 

= CO = R = say, 855 feet 



log. 1.934498 
log. 9.872321 



log. 1.806819 
log. 8.874938 



log. 2.931881 



2. In the case of a rectangular intersection, the solution is 
more simple. It is quite plain, from the figure, that — 

R 2 =(R — a) 2 +(R — 6) 2 , 
from which equation, 

R = a + b + \f2ab. 




FROGS AND SWITCHES. 



129 



Example. 

a = 40, 6 = 80. 

Then R = 40 + 80 + VM00 = 200. 

3. Cases of this kind are disposed of with great ease in the 
field hy means of the curve-protractor. 



XLVI. 



FROGS AND SWITCHES. 




TO FIND THE RADIUS OF A TURNOUT CURVE, THE FROG 
ANGLES, AND THE DISTANCES FROM THE TOE OF SWITCH 
TO THE FROG POINTS. 

1. Draw the figure as in margin, C being the centre of the 
turnout curve, CK parallel to main track, and OK, IE, LM, 
perpendicular to it. Call the angle of the frogs at O, F; that 
of the intermediate frog at I, 2 F'; the throw of the switch-rail 
for single turnout, D; its angle with main track, S; the gauge 
of the track, G; and radius of outer rail, R. 

2. Usually the length and throw of switch-rail and the 
angles of the frogs at O are given. In that case, to find R, F' 
and the distances LO, LI, reason thus: — 



130 FROGS AND SWITCHES. 

3. The angle HNW, between the line of the switch-rail pro- 
longed and a tangent to turnout curve at frog point O, = 
NOP — NHW = F — S. The angle NOL or NLO, be- 
tween chord and tangent, = half the intersection angle II N W 
= |(F — S). TheangleNOB = NOL + LOB. ButNOL 
is seen to be = £ (F — S), and NOB = F; then L OB = 
NOB-NOL = F-J(F-S) = i(F + S). The distance 
L O, from toe of switch to point of main frog, = L B -f- sin. 
LOB = (G-D)-r sin. £ (F + S). 

4. Again: the angle LCY = NLO =£ (F — S); LY = £ 
L0 = i(G-D)4- sin. * (F + S). LY -f- sin. LCY = 
LC; i.e., [J (G — D) -f- sin. i (F + S)| -f- sin. $ (F — S) 
= R. 

5. R may be found otherwise, as follows : — 

OK = OC cos. KOC = R cos. F; LM = L C cos. CLM = 
Rcos. S; LM-OK = LB; i.e., R (cos. S — cos. F) = (G — 
D). Hence R = (G — D) -f- (nat. cos. S — nat. cos. F). 

6. If R be known, to find F. This equation gives nat. cos. F 
= nat. cos. S-|(G-D)-rR]. 

7. To find the angle, 2 F', of the middle frog at I. 

IE = IP + PE or OK; i.e., R cos. F' = JG + R cos. F. 
Hence nat. cos. F' = nat. cos. F + (-J G -f-R). 

8. The angle L I Y, by similar reasoning to that used in rela- 
tion to LOB, is found to be = $ (F' + S). The distance L I, 
from toe of switch to point of middle frog, = L Y -f- sin. LI Y 
= (J G — D) -f- sin. i (F + S). 

The preceding formulas translate into the following — 

RULES FOR FROGS AND SWITCHES. 

9. To find the Angle of Switch-Bail with Main Track. 
Divide its throw, in decimals, by its length : the quotient 
will be the natural sine of the angle sought. 

10. To find the Distance from Toe of Switch to Point of 
Main Frog. 

Subtract the throw of switch-rail from the gauge of track, 
both in decimals; call the remainder a. Add together the 
angle of switch-rail with main track and the angle of the 
main frog; find the natural sine of half this sum, and call 
it b. Divide a by b: the quotient will be the distance 
sought. 



FROGS AND SWITCHES. 131 

11. To find the Radius of Outer Bail of Turnout Curve. 
Subtract the throw of switch-rail from the gauge of track, 
both in decimals; call the remainder a. Subtract the natural 
cosine of the main frog angle from the natural cosine of the 
switch-rail angle; call the remainder b. Divide a by b: the 
quotient will be radius. 

12. To find the Main Frog Angle, the Radius of the Outer 

Rail being known. 
Call the natural cosine of the switch-rail angle a. Subtract 
the throw of switch-rail from the gauge of track, both in deci- 
mals. Divide the remainder by radius; call the quotient b. 
Subtract b from a: the remainder will be the natural cosine of 
the main frog angle. 

13. To find the Angle of the Middle Frog, in the Case of 

a Double Turnout. 
Call the natural cosine of the main frog angle a. Divide 
half the gauge of track by the radius of outer rail of turnout 
curve ; call the quotient b. Add a and b together. Their sum 
is the natural cosine of half the middle frog angle. 

14. To find the Distance from Toe of Switch to Point of 

Middle Frog. 

Subtract the throw of switch-rail from half the gauge of 
track, both in decimals ; call the remainder a. Add together 
the switch-rail angle and half the middle frog angle. Find the 
natural sine of half this sum ; call said natural sine b. Divide 
a by b : the quotient will be the distance sought. 

15. The use of logarithms will be found convenient in work- 
ing these rules. 

Examples. 

16. Switch-rail, 18 feet; throw, 5 inches = 0.42 feet; frog 
angle, 6° 44'; gauge, 4.71 feet. 

Sin. S = 0.42 -=-18 = .02334 = sin. 1° 20'. 

LO = (G — D) -=- sin. J (F + S) = (4.71 — 0.42) -f- sin. 
3° 32' = 4.29 -=- 0.0616 = 69.64 feet. 

R = (G — D) -=- (nat. cos. S — nat. cos. F) = 4.29 -=- 0.00473 
= 907 feet. 

Nat. cos. F' = nat. cos. F + (| G -f- R) = 0.995 + (2.354 ~- 



232 FROGS AND SWITCHES. 

907) = 0.99759 = cos. 3° 58£'. Hence the angle of the middle 
frog = 2 F' = 7° 57'. 

L I = (i G — D) -f- sin. i (F' + S) = (2.354 — 0.42) ■»- sin. 
i (3° 58*' + 1° 20') = 1.934 -~ 0.0403 = 41.8 feet. 

17. In ordinary practice, frogs may be located with sufficient 
exactness by the following rules, deduced from the congruity 
of triangles. Great nicety in their location is not necessary. 
The important thing in practice is to lay the turnout curve so 
that the approach to the frog shall be fair and regular. How 
trackmen may do this without the use of instruments, in a 
very simple way, will be shown hereafter. Not that frogs may 
be set hap-hazard, and the approaches forced to fit: they 
ought to be nearly where they mathematically belong, and they 
can be thus placed by means of the rules subjoined. 

18. Let N stand for the number of the frog; 

L the length of switch-rail in feet; 
F the distance from toe of outer switch-rail to point 
of frog in feet. 
Then, for standard gauge, 4 feet 8| inches, straight switch- 
rail, and 5 inches throw of switch. 

8.6 L N 
L + 0.42 N* 

The above may be written roundly as a rule thus: — 
Multiply the length of switch-rail in feet by the number of 
the frog, and set down the product. Multiply that product by 
8£, and call the result A. Next add together the length of 
switch-rail in feet and two-fifths of the frog number; call the 
sum B. Then divide A by B, and the quotient will be the dis- 
tance in feet from toe of outer switch-rail to point of frog. 

Example. 
Switch-rail, 20 feet long; frog, No. 9. 

Length of switch-rail 20 

Multiplied by frog number .... 9 

Product . . 180 

Multiplied by 8£ 

1,530=- A. 
Length of switch-rail ...... 20 

Added to f frog No. 9 3.6 

23.6 = B. 



FROGS AND SWITCHES. 133 

A divided by B = 1,580 divided by 23.6 = 64,8 feet, the frog 
distance; say, 65 feet. 

19. If the switch-rail be curved, the formula would stand 
Jims: — 

8.6 LN 



F = 



L + 0.84 N 



Which may be made a written rule as follows : — 
Multiply the length of switch-rail in feet by the number of 
the frog, and their product by 8i; call the result A. Add 
together the length of switch-rail in feet and four-fifths of the 
frog number; call the sum B. Then divide A by B, jyid 
the quotient will be the distance from toe of outer switch-rail 
to point of frog in feet. 

20. The foregoing rules are applicable to turnouts from 
curves, as well as from straight lines. 

21. To find the radius of outer rail of a turnout curve from 
straight track. Data same as in previous rules for frogs; K 
the required radius in feet. 

8.6 L 2 N 2 
If the switch-rail be straight, R = f-n ' 



L 2 — 0.17 X' 2 

8.6 L 2 X 2 
If the switch-rail be curved, R = T 2 „~ ^ 2 . 

22. To find the radius of the outer rail of a turnout curve 
from curved track, proceed thus : — 

First find the radius as for a turnout from straight track by 
the preceding rule; call it, as before, R. Call the radius of the 
main track R 2 , and the required radius of turnout curve r. 

Then, if the turnout be towards the concave side of main 
track, — 

R 2 X R 
r " R 2 + R 

If the turnout be towards the convex side of main track, — 

R 3 X R 

r =. — . 

R. 2 — R 

More explicitly, in the first case, r is equal to the product of 
the other radii divided by their sum; and, in the second case, 
r is equal to the product of the other radii divided by their 
difference. 



134 FROGS AND SWITCHES. 

23. The angle of a frog is equal to 3,440' divided by the frog 
number. 

24. To find the frog distances and radii for a three-foot 
gauge, find them by the preceding rules for standard gauge, 
and take five-eighths of the result, using a switch-rail reduced 
\n like measure. 

For a metre gauge, take seven- tenths of the result, using a 
6witch-rail reduced in like measure. 

Or these radii and distances may be found from the appended 
tables for standard gauge by pro-rating as above. 

25. Three frog patterns are enough for general service. 
They should be so proportioned, that, taken in couples, the less 
may fit as middle frogs on double turnouts. Numbers 5J, 71, 
and 10£ make an excellent suit; numbers 5, 7, and 9| also 
answer very well. 

26. At the terminal stations, and about the shops of busy 
roads, patterns necessarily multiply. The better way in such 
cases is to plot the situation to a large scale, and to take the 
required distances and angles from the drawing. 






TURNOUT TABLE. 



135 



ljOU 

o 2. 

II 



P'P'P'I* 

° © Eft 
^CTQ ftr£ 

P /-, ^ 

^ 2. S. 5S" 



otOCO CO 

,_i m oo to 

<^COCO b 



*otO^ CO 

CO Cn p 



<StO^ 



^oCO^ 

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136 



TURNOUT TABLE. 



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TO LOCATE A TURNOUT . 



137 



XL VII. 

TO LOCATE A TURNOUT. 

1 . Let the heavy parallels in the figure represent the rails of 
the main track. 




2. Stick a pin or drive a spike at A, the toe of switch, at, 
a distance from the gauge side of the main-track rail equal to 
the throw of the switch-rail. Lay off the distances AC and 
AB (if a double turnout), taken from the foregoing tables, and 
place the frogs C and B, or mark those points. Stretch the 
cord from A to B, and from B to C. Mark the middle points 
of those stretches at H and P. Catch the cord at H with your 
forefinger, and pull it outwards until your finger, at E, lines 
with the switch-rail, and also with the right gauge side of frog 
B. Stick a pin at L, half-way between H and E. Let the 
cord spring in against L, so that it shall stretch straight from 
A to L, and from L to B. Opposite the middle points, V, of 
those stretches, stick pins on the outside at a distance from 
the cord equal to one-quarter of H L. In like manner, catch 
the cord at P, the point midway between B and C; stretch it 
to F, in line with the gauge sides of the frogs; and stick a pin 
at I, half-way between P and F. 



138 TO LOCATE A TURNOUT. 

3. Next lay off the proposed line of the near rail of the sidG 
track, X D. Mark the point G on that line where the range 
of the proper gauge side of frog C strikes it. Measure C G. 
Set off G D, equal to C G, along the side-track line, and drive 
a pin at D. Stretch the cord from C to D. Mark the middle 
point of it at K, and drive a pin at N, half-way between K and 
G. Stretch the cord from C to N, and from N to D. Stick 
pins outside the middle points, M and O, of those stretches at 
a distance from those points, M and O, equal to one-quarter of 

4. These three sets of pins will fix the line of one rail of the 
turnout. The corresponding rail of a double turnout can be 
laid off from them, if required, by symmetrical measurements. 

5. In the case of a single turnout, stretch the cord from the 
toe of switch, as above, to the point of frog, located by the 
foregoing tables ; catch it at the middle, and pull it outwards 
to a point in range with the line of the switch-rail in one 
direction, and the gauge side of frog in the other direction. 
Half-way between that point and the middle of the cord, when 
straight, stick a pin. Measure that half-way distance, and 
divide it by 4; call the quotient the " quarter-distance." 
Stretch the cord from the pin just set to the toe of switch in 
one direction, and to the point of frog in the other. Outside 
the middle points of these short stretches, lay off the " quarter- 
distance," as above found, and stick two other pins. These 
three pins will sufficiently mark the line of the outer rail of 
the turnout. 

6. The same methods will apply in practice to turnouts from 
curves. In the latter case, the distance C G is to be calculated 
as follows : — 

Multiply the distance Y D, between the nearest rails of the 
parallel tracks, by the number of the frog, taken from the fore- 
going table. Thus, on the full gauge, with a space between 
tracks of 7 feet and a No. 6 frog, the distance C G would be 7 
X 6 = 42 feet. Lay off C G, in range of the gauge side of the 
frog, and stick a pin at G. Measure out G D, equal to C G. 
and set another pin at D, making D Y the proper distance be- 
tween tracks. Then stretch the cord from D to C, and pro- 
ceed to stake off the curve CND, as above directed. 






CROSSINGS ON STRAIGHT LINES. 



139 



XLVIII. 

CROSSINGS ON STRAIGHT LINES 

1. Having located frogs B and C by the preceding methods, 
stretch the cord any convenient distance, C D, in the range of 



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the outer gauge side of the frog C. Set off E F parallel to 
C D, and distant the gauge-width from it. The intersection of 
said parallel at F with the near rail of the side track marks 
the spot for point of side-track frog; the curve F G, thence to 
toe of switch, corresponds to A C on the main track, and may 
be staked out in like manner. 



XLIX. 

CROSSINGS ON CURVES. 

1. Having located frogs B and C by the preceding methods, 
set off the width of gauge, C D, from point of frog C, and 
square to its outer gauge side. Stick a pin at D. 

2. Next calculate the distance D E to the point of side-track 
frog as follows: Subtract the gauge of track from the dis* 



140 



CROSSINGS ON CURVES. 



tance, H I, between the gauge sides of the nearest rails of the 
main and side tracks; multiply the remainder b}^ the number 




of the frog, taken from foregoing tables. The product will be 
the distance from D to the point of side-track frog at E. 



ELEVATION OF THE OUTER RAIL ON CURVES. 14j 

3. Suppose, for example, the gauge sides of the nearest rails 
of the main and side tracks are 6 feet 6 inches asunder; gauge 
of track, 4 feet 8| inches; frog, a No. 9. Reducing inches to 
decimals, we have then the distance between tracks 6.5 feet, 
less the gauge, 4.7 feet, = 1.8 feet; and 1.8 multiplied by 9, 
the number of the frog, gives 16.2 feet for the distance D E. 
The proper spring will be given to rail D E on the ground ; and 
curve E G, from frog to toe of side-track switch, will be staked 
off as directed in the section on turnouts. 



L. 
ELEVATION OF THE OUTER RAIL ON CURVES. 

1. Great precision in this adjustment is unattainable, owing 
to differences in the speed of trains and to the cost of track- 
maintenance, if it were attempted. 

2. Molesworth gives the following formula for determining 
the elevation of the outer rail with any gauge : — 

V = greatest velocity of trains in feet per second. 
G = gauge of railway in feet. 

C = length of chord whose 4niddle ordinate will give the 
required elevation. 

Then C = |Y ^ GT 

A modification of 
this formula gives the 
following approximate 
rules : — 

To fix the elevation 
of the outer rail on the standard gauge of 4 feet 8J- inches, 
multiply the speed of trains in miles per hour by 5, and divide 
the product by 3. This will give the length of tape, C, to 
stretch on the gauge side of the outer rail ; and the distance, e, 
from the middle of the tape to the gauge side of the rail, will 
be the proper elevation. 

For guage of one metre, = 3.28 feet, make C equal to one 
and one-third times the speed of trains in miles per hour. 

For 3-feet gauge, make C equal to one and one-fourth time 
the speed of trains in miles per hour. 




142 ELEVATION OF THE OUTER RAIL ON CURVES. 



TABLE OF ELEVATIONS OF OUTER RAIL ON CURVES. 

This table was formulated by the writer from Pennsylvania Rail Road 
practice as follows : 

(x^ speed in miles per hour + 1) X (by the degree of curve) = elevation of 
outer rail expressed in 8ths of an inch. 



o . 

W ° 


SPEED IN MILES PER HOUR. 


10 


20 


30 


40 


50 


60 








P 




VALUES IN EIGHTHS OP AN INCH. 




2° 


4 


6 


8 


10 


12 


14 


4° 


8 


12 


16 


20 


24 


28 


6° 


12 


18 


24 


30 


36 


42 


8° 


16 


24 


32 


40 


48 




10° 


20 


30 


40 


50 


. . 




12° 


24 


36 


48 








14° 


28 


42 










16° 


32 


48 











Note.— The limit of elevation of outer rail is 64 inches. 



TRACKMEN'S TABLE OF CURVES. 



llB 



LI. 



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DEGREE OF CURVE. 




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18 1-4 
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26 1-4 
29 

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34 

36 3-4 
39 1-4 
42 

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LENGTH OF CHORD IN FEET AND 

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TRACK. 


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DEGREE OF CURVZ. 



144 TRACKMEN'S TABLE OF CURVES. 



EXPLANATION OF THE FOEEGOING TABLE. 

Columns 1 and 10 give the degree of curve. 

The use of column 2, containing the deflection distances, 
may be illustrated thus : Suppose stakes 4, 5, and 6 to be miss* 
ing from a 3-degree curve, and that stakes 2 and 3 are still 
standing 100 feet apart. To replace the missing stakes, pro- 
ceed as follows: Measure 100 feet from 3 to A, and make a 
mark at A exactly in range with 2 and 3. Find, in column 2 
of the table, the deflection distance for a 3-degree curve, which 
is seen to be 5 feet 3 inches. Hold one end of the tape at A 




and, stretching 5 feet 3 inches towards 4, nearly square to the 
range A-3, make a scratch on the ground three or four feet 
long, swinging the tape around A as a centre. Next lay off 
100 feet from stake 3 to the scratch; where the end of that 
measurement strikes it, is the place for stake 4. By measuring 
100 feet out to B on the range 3-4, and proceeding in like 
manner, stake 5 may be set; and so on. 

3. If the centre line is already staked for track at points 100 
feet asunder, and the degree of curve is wanted, range out the 
straight line between stakes, as above, to A or B, and measure 
across from those marks to the neighboring location-stake. 
Suppose the distance B-5, for example, to be 8 feet 9 inches. 
Referring, then, to column 2 of the table, we find that deflec- 
tion distance to indicate a 5-degree curve. If the distance 






TRACKMEN'S TABLE OF CURVES. 



145 



proved to be 4 feet 4 inches, we should soon discover that that 
distance was about half-way between 3 feet 6 inches and 5 feet 
3 inches, the nearest numbers in the table corresponding 
respectively to a 2-degree and a 3-degree curve, and showing 
the located line to be a 2J-degree curve. 

4. Let A C B in the figure, which is drawn very much out of 
proportion in order to make the subject clear, represent the 
centre line of a curve. Suppose GH to be a chord 100 feet 
long, and G C or C H to be a chord 50 feet long. Then column 
3 in the table gives the distance, C D, from the middle of the 
100-feet chord to the rail, and column 4 gives the distance, 
E F, from the middle of the 50-feet chord to the rail, for the 
different degrees of curve. By the aid of these columns, pins 
can be set 25 feet apart on a curve where the location-stakes 
are 100 feet apart. Thus, for a 3-degree curve, C D is 8 inches, 




and EF 2 inches. If pins were wanted at the half-way marks 
N, their distance from the dotted short chords would be one^ 
quarter of E F. It must be an uncommon case, however, that 
calls for stakes closer together than 25 feet. 

5. Columns 5, 6, and 7 give the spring of rails of different 
lengths for the various degrees of curve. 

6. Columns 8 and 9 give figures for finding the degree of 
curve, by simple measurement of a straight line on the track, 
as follows : Suppose A C B and K I L to represent the rails of a 
curving track. From any point A, on the outer rail, sight 
lcross to a point B, on the same rail, along a line just touching 
the inner rail at I. Measure from A to B, and seek the dis- 
tance in column 8 or 9, according to the gauge of track. If 
the distance, for example, measured 232 feet on the full gauge, 
then the curve would be a 4-degree curve; if 249 feet, then it 
would indicate a 3|-degree curve, for the reason that the 



146 TRACKMEN'S TABLE OF CURVES. 

measured distance falls half-way between the distances corre- 
sponding to a 3-degree and a 4-degree curve respectively. 

7. The rate of curve can be found also very nearly by means 
of column 3. To do so, stretch a straight line, 100 feet long, 
between points on either rail ; for, though they seem very dif- 
ferent in the figure, the two rails of a track have practically 
the same curvature. Measure from the middle of the line 
across to the gauge side of the rail, and seek the measured 
distance in column 3: opposite to it, in column 1, will be 
found the degree of curve. 

8. If, in any case, the exact figures sought are not found in 
the table, take out the next figure less and the next greater. 
Subtract one from the other, and divide the remainder by 4. 
Acid the fourth part of the difference between them, thus 
determined, to the smaller number, and compare the sum with 
the number sought. If still too small, add another fourth 
part ; and so on until the distance or the degree is ascertained 
to within a quarter part. 

9. Suppose, for instance, a deflection distance measures 5 
feet 7 inches. The nearest tabular numbers are 5 feet 3 inches 
and 7 feet. Their difference is 21 inches, one-fourth of which 
is 5J inches. Adding h\ inches to the smaller number, 5 feet 
3 inches, gives 5 feet 8 \ inches, which indicates nearly enough 
a 3J-degree curve. Again: if a measurement of 175 feet n 
sought in column 9, the track is seen at once, without calcula- 
tion, to be a 4§-degree curve. 



TABLES 



TABLES OF THE TIMES OF CULMINATION AND 

OF ELONGATION OF THE POLE-STAB AND 

OF ITS AZIMUTH AT ELONGATION. 

These tables are designed to facilitate the determination of a 
meridian line and of the magnetic declination (variation of 
the compass) by simple instrumental meaus (p. 44). For this 
purpose the tables are sufficiently accurate. They will also 
be found useful when preparing for or laying out work for 
a more refined determination of the astronomical azimuth 
and for the measures of the value of an eye-piece micrometer. 



148 



TABLE I. 



MEAN LOCAL (ASTRONOMICAL) TIME, COUNTED FROM NOOIO 
AND FROM ZERO TO TWENTY-FOUR HOURS, OF THE 
CULMINATIONS AND ELONGATIONS OF POLARIS IN THE 
YEAR 1889. COMPUTED FOR LATITUDE 40° NORTH AND 
LONGITUDE 6 HOURS WEST FROM GREENWICH. 











1918 










Date. 


E. Elong. 


Upper Culm. 


W. 


Elong. 


LowerCulm. 






h. 


m. 


h. 


m. 


h. 


m. 


h. 


m. 


Jan. 


1 


18 


53.5 





48.7 


6 


43.5 


12 


46.8 


' 4 


15 


17 


48.2 


23 


43.4 


5 


38.6 


11 


51.5 


Feb. 


1 


16 


51.1 


22 


46.3 


4 


41.5 


10 


34.0 


' • 


15 


15 


53.8 


21 


49.0 


3 


46.2 


9. 


47.0 


Marcl 


i 1 


15 


00.6 


20 


55.8 


2 


51.0 


8 


53.8 


' ' 


15 


13 


55.4 


19 


50.6 


1 


45.8 


7 


50.6 


April 


1 


12 


58.5 


18 


53.7 





48.9 


6 


51.7 


' • 


15 


12 


03.4 


17 


58.6 


23 


53.8 


5 


56.6 


May 


1 


10 


56.7 


16 


51.9 


22 


47.1 


4 


49.9 


' ' 


15 


9 


59.7 


15 


54.9 


21 


50.1 


3 


42.9 


June 


1 


8 


55.2 


14 


50.4 


20 


59.5 


2 


48.4 


' ' 


15 


8 


00.3 


13 


55.5 


19 


50.7 


1 


53.; 


July 


1 


6 


57.3 


12 


52.5 


18 


47.7 





40.5 


1 ' 


15 


6 


01.9 


11 


57.1 


17 


52.3 


23 


55.1 


Aug. 


1 


4 


56.4 


10 


51.6 


16 


46.8 


22 


49.6 


' ' 


15 


4 


01.8 


9 


57.0 


15 


52.2 


21 


55.0 


Sept. 


1 


2 


54.8 


8 


50.0 


14 


55.3 


20 


58.0 


' ' 


15 


2 


00.1 


7 


55.3 


13 


50.5 


19 


53.3 


Oct. 


1 





57.4 


6 


52.6 


12 


47.8 


18 


50.6 


' ' 


15 


23 


46.7 


5 


41.9 


11 


37.1 


17 


39.9 


Nov. 


1 


22 


59.5 


4 


54.7 


10 


49.9 


16 


52.7 


' ' 


15 


22 


00.5 


3 


55.7 


9 


50.9 


15 


53.7 


Dec. 


1 


21 


01.4 


♦2 


56.6 


8 


51.8 


14 


54.6 




15 


20 


02.2 


1 


57.4 


7 


52.6 


13 


55.4 



To refer the tabular times to any year subsequent to the 
tabular year (1918) add m .33 for every year. 

To refer the tabular times, corrected as above, to any year 
iu a quadrennium, observe the following rules: 

For the first year after a leap-year the table is correct. 

For the second year after a leap-year add m .9 to the tabular 
value. 

For the third year after a leap-year add l m .7 to the tabular 
value. 

For leap year and before MarcL 1 add 2 m .6 to the tabular 
value. 

For leap-year from and after March 1 subtract l m .2 from the 
tabular value. * 



CULMINATIONS AND ELONGATIONS OF POLARIS. 149 



To refer to any calendar day other than the 1st and 15th of 
each month, subtract 3 m .94 for every day between it and the 
preceding tabular day, or add 3 m .94 for every day between it 
and the succeeding tabular day. 

The longitude correction will amount to m .16 for each hour. 

To refer to any other than the tabular latitude, and between 
the limits of 25° and 50° North, add to the time of west elonga- 
tion m .13 for every degree South of 40° and subtract from the 
time of west elongation m .18 for every degree North of 40°. 
Reverse these signs for corrections to times of east elongation. 

Observe that the year 1900 is not a leap-year, and. this must 
be kept in view when dealing with dates from and after 
March 1 of that year. The 20th century begins after the ex- 
piration of Dec. 31, 1900. 

The deduced tabular times may be relied on to have no 
greater error than ± m .3. 

Table II. below Lat. 24° is abridged from a table for each 
degree of latitude between 25° and 50° North, computed for 
this book by Mr. C. A. Schott, Asst. Supt. of the U. S. C. and 
G. Survey, with the mean declination of Polaris for each 
year. A closer result will be had by applying to the tabular 
values the following correction, which depends on the differ- 
ence of the mean and the apparent places of the star : 



For 
Middle of 


Lat. 25° 


Lat. 40° 


Lat. 50° 


For 

Middle of 


Lat. 25° 


Lat. 40° 


Lat. 50° 


Jan. 


-0'.3 


- 0'.4 


- 0'.4 


July 


+ / .2 


+ 0'.3 


+ , .3 


Feb. 


-0.3 


-0.3 


- 0.4 


Aug. 


+ 0.1 


+ 0.1 


+ 0.2 


March 


-0.1 


- 0.2 


- 0.2 


Sept. 


0.0 


- 0.1 


- 0.1 


April 


0.0 


0.0 


0.0 


Oct. 


- 0.2 


-0.3 


-0.3 


May 


+ 0.2 


+ 0.2 


+ 0.2 


Nov. 


- 0.5 


-0.6 


- 0.7 


June 


+ 0.3 


+ 0.3 


+ 0.3 


Dec. 


-0.6 


-0.8 


-0.9 



The deduced tabular azimuth, counted from the North, 
may generally be depended upon with no greater error than 
±0'.2. 

In making the computation the mean places of Polaris were 
first accurately deduced from New comb's Catalogue of 1098 
standard clock and zodiacal stars, Washington, 1881, for five 
equidistant epochs. From these fundamental places those for 
each year were readily found by interpolation. 

Azimuth for latitudes less than 25° was reckoned by the 
author from the data for that degree. 



150 



AZIMUTH OF POLARIS AT ELONGATION. 






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TABLES. 



151 



TABLE III. 



ROODS AND PERCHES IN" DECIMAL PARTS OF AN ACRE. 
One Acre =4 Roods = 160 Perches = 4,840 Square Yards = 43,560 Square Feet. 



(A 
u 
X 
u 
« 

u 


Roods. 


t/5 

w 

X 
u 

K 
W 

21 




Roods. 







1 


2 


3 





1 


2 


3 


o 


.000 


.250 


.500 


75o 


•131 


381 


631 


881 


I 


.006 


.256 


.506 


756 


22 


•*37 


387 


637 


887 


2 


.012 


.262 


.512 


762 


23 


.144 


394 


644 


894 


3 


.019 


.269 


•519 


769 


24 


.150 


400 


650 


900 


4 


.025 


•275 


•525 


775 


25 


.156 


406 


656 


906 


5 


.031 


.281 


•53i 


781 


26 


.162 


412 


662 


912 


6 


•037 


.287 


•537 


787 


27 


.169 


419 


669 


919 


7 


.044 


.294 


•544 


794 


28 


•175 


425 


675 


925 


8 


.050 


.300 


•55o 


800 


29 


.181 


43i 


681 


93* 


9 


.056 


.306 


.556 


806 


30 


.187 


437 


687 


937 


ro 


.062 


.312 


.562 


812 


31 


.194 


444 


694 


944 


it 


.069 


.319 


•569 


819 


32 


.200 


450 


700 


95o 


12 


•075 


•325 


•575 


825 


33 


.206 


456 


706 


956 


J 3 


.081 


•33i 


.581 


831 


34 


.212 


462 


712 


962 


*4 


.087 


•337 


.587 


837 


35 


.219 


469 


719 


969 


!5 


.094 


•344 


•594 


844 


3b 


.225 


475 


7 2 5 


975 


16 


.100 


.350 


.600 


850 


37 


.231 


481 


73i 


981 


i7 


.106 


•356 


.606 


856 


38 


•237 


487 


737 


987 


18 


.112 


.362 


.612 


862 


39 


•244 


494 


744 


994 


*9 


.119 


•369 


.619 


869 


40 


• 250 


500 


75o 1 


000 


20 


.125 


•375 


.625 


875 



















TABLE IV. 

DECIMALS OF AN ACRE IN ONE CHAIN LENGTH OF 100 FEET, 
AND OF VARIOUS WIDTHS. 



Width in 
Rods. 


Decimals of an 

Acre per 100 

Feet. 


Acres per 
Mile. 


Width in 
Rods. 


Decimals of an 

Acre per 1 00 

Feet. 


Acres per 
Mile. 


1 

2 

2^ 

3 

3^ 

4 

A l A 

5 


.018939 
.037879 
.056818 

•075757 
.094697 
.113636 
•132576 
■151515 
.170454 
.189394 


1 
2 
3 
4 
5 
6 

7 

8 

9 
10 


$% 

6 

6% 

7 , 

7% 

8 

sy 2 
9 

9 X A 
10 


.208333 
.227273 
.246212 
.265151 
.284091 
.303030 
.321970 
.340909 
.359848 
.378788 


11 
12 
13 
14 
15 
16 

17 
18 

19 
20 



152 



TABLES. 



TABLE V. 

ACRES, ROODS, AND PERCHES IN SQUARE FEET. 



Acres. 


Square Feet. 


Roods. 


Square Feet. 


Perches. 


Square Feet. 


i 


• 4356o 


1 


10890 


17 


4628.25 


2 


87120 


2 


21780 


18 


4900.50 


3 


130680 


3 


32670 


19 


5172.75 . 


4 


174240 


4 


4356o 


20 


5445- 00 


5 
6 

7 
8 


217800 
261360 
304920 
348480 






21 
22 
23 


57I7-75 
5989-50 
6261.75 
6534.00 


Perches. 


Square Feet. 






24 


9 


392040 


1 


272.25 


25 


6806.25 


IO 


435600 


2 


544-So 


26 


7078.50 


ii 


479160 


3 


816.75 


27 


7350.75 


12 


522720 


4 


1089.00 


28 


7623.00 


13 


566280 


5 


1361.25 


29 


7895-25 


14 


609840 


6 


1633.50 


30 


8167.50 


15 


653400 


7 


1905-75 


31 


8439-75 


16 


696960 


8 


2178.00 


32 


8712.00 


17 


740520 


9 


2450.25 


33 


8984.25 


18 


784080 


10 


2722.50 


34 


9256.50 


19 


827640 


11 


2994-75 


35 


9528.75 


20 


871200 


12 


3267.00 


36 


9801 .00 


21 


914760 


13 


3539-25 


37 


10073.25 


22 


958320 


14 


3811.50 


38 


10345.50 






15 


4083.75 


39 


10617.75 






16 


4356.00 


40 


10890.00 





TABLE VL 

CIRCULAR ARCS TO RADIUS OF 1. 





Degrees. 


Minutes. 


Seconds. 


• 
1 
2 
3 
4 
5 
6 

7 
8 

9 


.01745329 
.03490658 
.05235988 
.06981317 
.08726646 
.10471975 
.12217305 
.13962634 
.15707963 


/ 
1 
2 
3 
4 
5 
6 

7 
8 

9 


.00029089 
.00058178 
.00087266 
.00116355 
.00145444 
.00174533 
.00203622 
.00232711 
.00261799 


// 

1 
2 
3 
4 
5 
6 

7 
8 

9 


.00000485 
.00000970 
.00001454 
.00001939 
.00002424 
.00002909 
.00003394 
.00003878 
.00004363 





TABLES. 



153 



TABLE VIL 

FEET IN DECIMALS OF A MILE. 





Feet. 


Decimals of a Mile. 


I 


o.o 00189394 


2 


0.0 00378798 


3 


0.0 00568182 


4 


0.0 00757576 


5 


0.0 00946970 


6 


0.0 01 136364 


7 


0.0 1 3 2 5 7 5 8 


8 


0.0 1 5 1 5 1 5 2 


9 


0.0 01704546 





TABLE YIII. 

INCHES REDUCED TO DECIMAL PARTS OF A FOOT. 





In. 





X 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


In. 


1? 


.0000 


•0833 


.1667 


.2500 


•3333 


.4167 


.5000 


.5833 


.6667 


.7500 


.8333 


.9167 





1* 


.0052 


.0855 


.1719 


.2552 


.3385 


.4219 


.5052 


•5885 


.6719 


•7552 


.8385 


•9219 


tV 


' \ 


.0104 


.0938 


.1771 


.2604 


.3438 


.4271 


.5104 


.5938 


.6771 


.7604 


.8438 


.9271 


1 


\ A 


.0156 


.0990 


.1823 


.2656 


•349° 


•4323 


•5156 


•599o 


.6823 


.7656 


.8490 


•9323 


A 


! \ 


.0208 


.1042 


.1875 


.2708 


•3542 


•4375 


.5208 


.6042 


.6875 


.7708 


.8542 


•9375 


i 


1 1% 


.0260 


.1094 


.1927 


.2760 


•3594 


■4427 


.5260 


.6094 


.6927 


.7760 


.8594 


•9427 


5 

16 


; 1 


•03I3 


.1146 


.1979 


.2813 


.3646 


•4479 


•53 1 3 


.6146 


.6979 


.7813 


.8646 


•9479 


3 


A 


•0365 


.1198 


.2031 


.2865 


.3698 


•453i 


•5365 


.6198 


•7031 


.7865 


.8698 


•953i 


tV 


i 


.0417 


.1250 


.2083 


.2917 


•375o 


.4583 


•54i7 


.6250 


•7083 


•79i7 


.8750 


.9583 


4 


& 


.0469 


.1302 


.2135 


.2969 


.3802 


•4635 


•5469 


.6302 


•7i35 


.7969 


.8802 


•9635 


9 
16 


* 


.0521 


•1354 


.2188 


.3021 


.3854 


.4688 


•5521 


•6354 


.7188 


.8021 


^8854 


.9688 


5 
8" 


.XI 
1 6 


•°573 


.1406 


.2240 


.3073 


.3906 


.4740 


•5573 


.6406 


.7240 


.8073 


.8906 


.9740 


11 
16 


\ 


.0625 


.1458 


.2292 


.3125 


.3958 


•4792 


•5625 


.6458 


.7292 


.8125 


.8958 


•9792 


,* 


T§ 


0677 


.1510 


•2344 


.3177 


.4010 


.4844 


•5677 


.6510 


•7344 


.8177 


.9010 


■9844 


if i 


7 

8 


.QT2Q 


•1563 


.2396 


.3229 


.4063 


.4896 


•5729 


•6563 


•7396 


.8229 


.9063 


.9896 


7 

"8" 1 


16 
16 


.O781 


1615 


.2448 


.3281 


•4"5 


.4948 


.5781 


.6615 


.7448 


.8281 


-^S 


•9948 


— 1 



TABLE IX. 

RADII AND THEIR LOGARITHMS, MIDDLE ORDI- 
NATES, AND DEFLECTION DISTANCES. 

Note. — This table applies to chords of 100-feet units. To 
use it for chords of 20-metre units, divide the degree of the 
proposed metric curve by 2 and find the quotient in the first 
column of the table; or multiply any tabular degree of curve 
by 2 for the degree of the proposed metric curve. Then one- 
tenth of the radius opposite the tabular degree thus found or 
assumed will give very nearly the true radius in metres of the 
proposed metric curve, and four-tentlis of the corresponding 
figures in the last three columns will give very nearly the 
values of those functions in metric measure. Thus, if a 3° 
metric curve be proposed, we take one-tenth of the radius 
opposite the -tabular -degree 1° 30', or 382 metres, and find the 
values of the functions in the last three columns to be .1308, 
1.048, and .524. Resulting errors may be neglected except in 
work of extraordinary accuracy. 



156 



RADII AND THEIR LOGARITHMS. 



Degree 




Logarithm 


Arithmetical 


Middle 


Deflec- 


Tangen- 


of 


Radius. 


of 


Comple- 


Ordinate, 


tion Dis- 


tial Dis- 


j Curve. 




Radius. 


ment. 


Chord 
100 Feet. 


tance. 


tance. 


o / 
5 


68754.9 


4-837304 


5.162696 


.018 


•i45 


•073 


IO 


34377-5 


4-536274 


5.463726 


.036 


.291 


•H5 


15 


22918.3 


4.360182 


5.639818 


•055 


•436 


.218 


20 


17188.8 


4.235246 


5-764754 


•073 


.582 


.291 


25 


i375 I -o 


4-I38335 


5.861665 


.091 


•727 


•364 


30 


11459.2 


4-o59i54 


5.940846 


.109 


•873 


•436 


35 


9822.2 


3.992209 


6.007791 


.127 


1.02 


•509 


40 


8594.4 


3-9342I5 


6.065785 


•145 


1. 16 


.582 


45 


7 6 39-5 


3.883066 


6.116934 


.164 


i-3i 


•654 


50 


6875.5 


3-837304 


6.162696 


.182 


i-45 


.727 


55 


6250.5 


3-7959*4 


6 . 204086 


.200 


1.60 


.800 


1 


5729.6 


3.758128 


6.241872 


.218 


i-75 


.873 


5 


5288.9 


3-723365 


6.276635 


.236 


1.89 


•945 


10 


4911.1 


3.691179 


6.308821 


•255 


2.04 


1.02 


15 


4583-7 


3.661216 


6.338784 


•273 


2.18 


1.09 


20 


4297-3 


3-633I95 


6.366805 


.291 


2-33 


1. 16 


25 


4044.5 


3.606864 


6.393136 


•309 


2-47 


1.24 


30 


3819.8 


3.582041 


6.417959 


•327 


2.62 


i-3i 


35 


3618.8 


3-558565 


6-44H35 


•345 


2.76 


1-38 


40 


3437-9 


3-536293 


6.463707 


•364 


2.91 


i-45 


45 


3274.2 


3.515106 


6.484894 


.382 


3-°5 


*-53 


50 


3125.4 


3.494906 


6.505094 


.400 


3.20 


1 .60 


55 


2989.5 


3-475599 


6.524401 


.418 


3-34 


1.67 


2 


2864.9 


3-457"4 


6.542886 


.436 


3-49 


1.74 


5 


2750.3 


3-43938o 


6.560620 


•455 


3-64 


1.82 


10 


2644.6 


3-422359 


6.577641 


•473 


3.78 


1.89 


15 


2546.6 


3.405961 


6.594039 


.491 


3-93 


1.96 


20 


2455-7 


3-39 OI 75 


6.609825 


•509 


4.07 


2.04 


25 


2371.0 


3-374932 


6.625068 


•527 


4.22 


2. 11 


30 


2292.0 


3.360215 


6.639785 


•545 


4-36 


2.18 


35 


2218. 1 


3-345982 


6.654018 


.564 


4-5i 


2.25 


40 


2148.8 


3-332196 


6.667804 


.582 


4-65 


2-33 


45 


2083.7 


3-318835 


6.681165 


.600 


4.80 


2.40 


50 


2022.4 


3.305867 


6.694133 


.618 


4-94 


2.47 


55 


1964.6 


3-293274 


6.706726 


.636 


5-o9 


2.54 


3 


1910.1 


3.281056 


6.718944 


.655 


5-23 


2.62 


5 


1858.5 


3.269163' 


6.730837 


•673 


5-38 


2.69 


10 


1809.6 


3-257584 


6.742416 


.691 


5-53 


2.76 


15 


1763.2 


3.246301 


6.753699 


.709 


5-67 


2.84 


20 


1719.1 


3-23530I 


6.764699 


.727 


5-82 


2.91 


25 


1677.2 


3-224585 


6.775415 


•745 


5-96 


2.98 


30 


1637.3 


3.214129 


6.785871 


•764 


6. 11 


3-05 


35 


1599.2 


3.203902 


6.796098 


.782 


6.25 


3-i3 


40 


1562.9 


3-I9393I 


6 . 806069 


.800 


6.40 


3.20 


45 


1528.2 


3.184180 


6.815819 


.818 


6.54 


3 2 7 


50 


1495.0 


3. 1 74641 


6.825359 


.836 


6.69 


3-34 


55 


1463.2 


3-165303 


6.834607 


.855 


6.83 


3-42 


4 


1432.7 


3-i56i55 


6.843845 


.873 


6.98 


3-49 


5 


1403-5 


3-147212 


6.852788 


.891 


7.12 


3-56 


10 


J375-4 


3.138429 


6.861571 


•909 


7.27 


3-63 


15 


1348.4 


3.129819 


6.870181 


.927 


7.42 


3-7 1 


20 


1322.5 


3-121395 


6.878605 


•945 


7-56 


3.78 


25 


1297.6 


3-U3MI 


6.886859 


•964 


7.71 


3.85 


30 


1273.6 


3- I0 5033 


6.894967 


.982 


7.85 


3-93 


35 


1250.4 


3.097048 


6.902952 


1. 00 


8.00 


4.00 


1 4° 

1 


1228. 1 


3-089233 


6.910767 


1.02 


8.14 


4.07 


L — .... . .— — . — 



BAD II AND THEIR LOGARITHMS. 



157 





Degree 




Logarithm 


Arithmetical 


Middle 
Ordinate, 

Chord 
100 Feet. 


Deflec- 


Tangen- 


of 


Radius. 


of 


Comple- 


tion Dis- 


tial Dis- 


Curve. 




Radius. 


ment. 


tance. 


tance. 


o / 

4 45 


1206.6 


3.081563 


6.918437 


1.04 


8.29 


4.14 


50 


1185.8 


3. 07401 1 


6.925989 


1.05 


8.43 


4.22 


55 


1165.7 


3.066587 


6.933413 


1.07 


8.58 


4.29 


5 


1146.3 


3.059299 


6.940701 


1.09 


8.72 


4-36 


5 


1127.5 


3-052117 


6.947883 


1. 11 


8.87 


4-43 


10 


1 109. 3 


3.045050 


6.954950 


1. 13 


9.01 


4-5i 


15 


1091.7 


3.038103 


6.961897 


I-I5 


9.16 


4.58 


20 


1074.7 


3.031287 


6.968713 


1. 16 


9-3o 


4.65 


25 


1058.2 


3.024568 


6.975432 


1. 18 


9-45 


4.72 


30 


1042. 1 


3.017910 


• 6.982090 


1.20 


9.60 


4.80 


35 


1026.6 


5.011401 


6.988599 


1.22 


9-74 


4.87 


40 


ion. 5 


3.004967 


6.995033 


1.24 


9.89 


4.94 


45 


996.9 


2.998652 


7.001348 


1.25 


10. 


5.02 


50 


982.6 


2.992377 


7.007623 


1.27 


10.2 


5-09 


55 


968.8 


2.986234 


7.013766 


1.29 


10.3 


5.16 


6 


955-4 


2.980185 


7.019815 


1. 3i 


10.5 


5-23 


5 


942-3 


2.974189 


7.025811 


1-33 


10.6 


5-3i 


10 


929.6 


2.968296 


7.031704 


i-35 


10.8 


5.38 


x 5 


917.2 


2.962464 


7-037536 


1.36 


10.9 


5-45 


20 


905.1 


2.956697 


7.043303 


1.38 


11. 


5-52 


25 


893-4 


2.951046 


7.048954 


1.40 


11. 2 


5.60 


30 


882.0 


2.945469 


7-o5453i 


1.42 


u-3 


5.67 


35 


870.8 


2.939918 


7.060082 


1.44 


n-5 


5-74 


40 


859-9 


2.934448 


7.065552 


i-45 


11. 6 


5-8i 


45 


849-3 


2.929061 


7.070939 


1.47 


11. 8 


5.89 


50 


839.0 


2.923762 


7.076238 


1.49 


11. 9 


5-9<> 


55 


828.9 


2.918502 


7^082498 


i-5i 


12. 1 


6.03 


7 


819.0 


2.913284 


7.086716 


i-53 


12.2 


6.10 j 


5 


809.4 


2.908163 


7.091837 


i-55 


12.3 


6.18 


10 


800.0 


2.903090 


7.096910 


1.56 


12.5 


6.25 


15 


790.8 


2.898067 


7- IOI 933 


1.58 


12.6 


6.32 


20 


781.8 


2.893096 


7.106904 


1.60 


12.8 


6-39 


25 


773-1 


2.888236 


7.111764 


1.62 


12.9 


6.47 


30 


764-5 


2.883377 


7.116623 


1.64 


i3-i 


6.54 


35 


756.1 


2.878579 


7.121421 


1.65 


13.2 


6.61 


40 


747-9 


2.873844 


7.126156 


1.67 


13-4 


6.68 


45 


739-9 


2.869173 


7.130827 


1.69 


13-5 


6.76 


50 


732.0 


2.864511 


7.135489 


1. 71 


13-7 


6.83 


55 


7 2 4-3 


2.859918 


7.140082 


1-73 


13.8 


6.90 


8 


716.8 


2.855398 


7.144602 


i-75 


14.0 


6.98 


5 


709.4 


2.850891 


7.149109 


1.76 


14. 1 


7-05 


10 


702.2 


2.846461 


7-!53539 


1.78 


14.2 


7.12 


15 


695.1 


2 . 842047 


7- x 57953 


1.80 


14.4 


7.19 


20 


688.2 


2-837715 


7.162285 


1.82 


14-5 


7.27 


25 


681.3 


2.833338 


7.166662 


1.84 


14.7 


7-34 


30 


674.7 


2.829111 


7 . 1 70889 


1.85 


14.8 


7.41 


35 


668.1 


2.824841 


7- I 75i59 


1.87 


15.0 


7.48 


40 


661.7 


2.820661 


7-*79339 


1.89 


15. 1 


7-56 


45 


655-4 


2.816506 


7.183494 


1. 91 


15.3 


7-63 


50 


649-3 


2.812445 


7- l8 7555 


i-93 


15.4 


7.70 


55 


643.2 


2.808346 


7.181654 


i-95 


15-5 


7-77 


9 


637-3 


2.804344 


7.195656 


1.96 


15-7 


7.85 


5 


631.4 


2.800305 


7.199695 


1.98 


15.8 


7.92 


10 


625.7 


2.796366 


7.203634 


2.00 


16.0 


7-99 
8.06 


15 


620.1 


2.792462 


7.207538 


2.02 


16. 1 





158 



RADII AND THEIR LOGARITHMS. 





Degree 




Logarithm 


Arithmetical 


Middle 


Deflec- 


Tangen- 


of 


Radius. 


of 


Comple- 


Ordinate, 


tion Dis- 


tial Dis- 


Curve. 




Radius. 


ment. 


Chord 
100 Feet. 


tance. 


tance. 


o / 
9 20 


614.6 


2.788593 


7.211407 


2.04 


16.3 


8.14 


25 


609.1 


2.784689 


7-2153" 


2.06 


16.4 


8.21 


30 


603.8 


2.780893 


7.219107 


2.07 


16.6 


8.28 


35 


598.6 


2.777137 


7.222863 


2.09 


16.7 


8.35 


40 


593-4 


•2.773348 


7.226652 


2. 11 


16.8 


8-43 


45 


588.4 


2.769673 


7.230327 


2.13 


17.0 


8.50 


50 


583-4 


2.765966 


7. 2 34i34 


2.15 


17. 1 


8.57 


55 


578.5 


2.762303 


7.237697 


2.16 


17-3 


8.64 


10 


573-7 


2.758685 


7- 2 4*3 1 5 


2.18 


17.4 


8.72 


10 


564-3 


2.751510 


7.248490 


2.22 


17.7 


8.86 


20 


555-2 


2-744449 


7.25555I 


2.26 


18.0 


9.00 


30 


546.4 


2. 7375" 


7.262489 


2.29 


18.3 


9-15 


40 


537-9 


2.730702 


7.269298 


2-33 


18.6 


9.30 


50 


529.7 


2.724030 


7.275970 


2.36 


18.9 


9-44 


11 


521.7 


2.717421 


7.282579 


2.40 


19.2 


9.58 


10 


5I3-9 


2.710879 


7.289121 


2.44 


19-5 


9-73 


20 


506.4 


2.704494 


7.295506 


2-47 


19.7 


9.87 


30 


499.1 


2.698188 


7.301812 


2.51 


20.0 


10. 


40 


492.0 


2.691965 


7.308035 


2-55 


20.3 


10.2 


50 


485.1 


2.685831 


7.314169 


2.58 


20.6 


10.3 


13 


478.3 


2.679700 


7.320300 


2.62 


20.9 


10.4 


10 


471.8 


2.673758 


7.326242 


2.66 


21.2 


10.6 


20 


465-5 


2.667920 


7.332080 


2.69 


21.5 


10.7 


30 


459-3 


2 . 662096 


7.337904 


2.73 


21.8 


10.9 


40 


453-3 


2.656386 


7-3436i4 


2.77 


22.1 


•11. 


5° 


447-4 


2.650696 


7.349304 


2.80 


22.4 


11. 2 


13 


441.7 


2.645127 


7-354873 


2.84 


22.6 


"•3 


10 


436.1 


2.639586 


7.360414 


2.88 


22.9 


"•5 


20 


430.7 


2.634175 


7-365825 


2.91 


23.2 


11. 6 


30 


425.4 


2.628797 


7.371203 


2-95 


23-5 


11.7 


40 


420.2 


2.623456 


7-376544 


2.98 


23.8 


11. 9 


50 


415.2 


2.618257 


7-38i743 


3.02 


24.1 


12.0 


14 


410.3 


2.613102 


7.386898 


3.06 


24.4 


12.2 


10 


405-5 


2.607991 


7.392009 


3-09 


24.7 


12.3 


20 


400.8 


2.602928 


7.397072 


3.i3 


25.0 


12.5 


30 


396.2 


2.597914 


7.402086 


3-*7 


25.2 


12.6 


40 


39 x -7 


2.592954 


7.407046 


3.20 


25-5 


12.8 


50 


387.3 


2.588047 


7.4"953 


3-24 


25.8 


12.9 


15 


383-1 


2.583312 


7.416688 


3.28 


26.1 


13.0 


10 


378.9 


2.578525 


7.421475 


3-3* 


26.4 


13.2 


20 


374-8 


2.573800 


7.426200 


3-35 


26.7 


13.3 


30 


37o.8 


2.569140 


7 . 430860 


3-39 


27.0 


13-5 


40 


366.9 


2.564548 


7.435452 


3-42 


27-3 


13.6 


50 


363.0 


2.559907 


7.440093 


3-46 


27-5 


13-8 


16 


359-3 


2.555457 


7-444543 


3-5o 


27.8 


13-9 


10 


355-6 


2.550962 


7.449038 


3-53 


28.1 


14. 1 


20 


352.o 


2.546543 


7-453457 


3-57 


28.4 


14.2 


30 


348.4 


2.542078 


7.457922 


; 3-6i 


28.7 


H-3 


40 


345-o 


2-537819 


7.462181 


3-64 


29.0 


14-5 


50 


341.6 


2.5335i8 


7.466482 


3.68 


29-3 


14.6 


17 


338.3 


2.529302 


7.470698 


3.72 


29.6 


14.8 


10 


335.o 


2.525045 


7-474955 


3-75 


29.9 


14.9 



RADII A2TD THEIR LOGARITHMS. 



159 





Degree 
of 


Radius. 


Logarithm 
of 


Arithmetical 
Comple- 


Middle 
Ordinate, 


Deflec- 
tion Dis- 


Tangen- 
tial Dis- 


Curve. 




Radius. 


ment. 


Chord 
100 Feet. 


tance. 


tance. 


o / 
17 20 

30 
40 


331.8 
328.7 
325-6 


2.520876 
2.516800 
2.512684 


7.479124 
7.483200 
7.487316 


3-79 
3.82 
3-86 


30.1 
30.4 
30.7 


i5-i 
15.2 
15-4 


50 


322.6 


2 . 508664 


7-49J336 


3.90 


31.0 


15-5 


18 

10 


319.6 
316.7 


2.504607 
2.500648 


7-495393 
7.499352 


3-93 

3-97 


31-3 
31.6 


15.6 
15.8 


20 


3I3-9 


2.496791 


7.503209 


4.01 


3i.9 


15.9 


30 
40 


311. 1 
308.3 


2.492900 
2.488974 


7.507100 
7.511026 


4.04 
4.08 


32.1 
32.4 


16. 1 
16.2 


50 


3°5-6 


2.485153 


7-514847 


4.12 


32.7 


16.4 


19 


302.9 


2.481299 


7.518701 


4.i5 


33-o 


16.5 


10 
20 

30 


3oo-3 
297.8 
295.2 


2-477555 
2.473925 
2.470116 


7.522445 
7.526075 
7.529884 


4.19 
4-23 
4.26 


33-3 
33-6 
33-9 


16.6 
16.8 
16.9 


40 


292.8 


2.466571 


7-533429 


4-30 


34-2 


17. 1 


50 


290.3 


2.462847 


7-537153 


4-34 


34-4 


17.2 


20 


287.9 


2.459242 


7-540758 


4-37 


34-7 


17.4 



TABLE X.— (See p. 160.) 
FOR USE WITH A 20-METRE CHAIN. 

Engineers accustomed tojhinking their degree of curvature 
with reference to the 100-ft. chain may find it convenient to 
remember that the degree of curvature, if a 20-metre chain be 
used, is approximately two-thirds of the foregoing. Thus a 
3° metric curve would be about equivalent to a 4J C curve laid 
out with the 100-ft. chain. 

A 20-metre chain =65.618 feet; a 100-ft. chain = 1.524 
chains of 20 metres each, one foot being equal to 0.3048 of a 
metre, and a metre equal to 3.2809 feet. 

If a metric curve is to be retraced with a 100-ft. chain, the 
exact degree of curvature should be ascertained with reference 
to the radius in feet, as set forth in Art. XVIII. 

It is convenient to mark stakes with the even numbers, 
2, 4, 6, etc , when using the 20-metre chain, distance being 
thus recorded in tens of metres. 



160 



METRIC CURVE TABLE. 



Degree 


Radius 


Loga- Arithme- Mi 


d.Ord 


Deflec- 


Tangen- 


of 


in 


rithm of ticalCom- Chord 


tion Dis- 


tial Dis- 


Curve. 


Metres. 


Radius, plement. 20 


Metres 


tance. 


tance. 


TO 


6875.50. 


3-837304 


6.162696 


.0076 


.0582 


.0291 


20 


3437-75 


3-536274 


6.463726 


.0144 


. 1 164 


.0582 


30 


2291.84 


3.360184 


6.639816 


.0218 


•1745 


•0873 


40 


1718.88 


3-235246 


6.764754 


.0290 


.2327 


.1164 


* 5 ° 


1375- 11 


3.138338 


6.861663 


.0363 


.2909 


• 1454 


1 


"45-93 


3-059158 


6.940842 


.0437 


•3491 


•1745 


10 


982.23 


2.992213 


7.007787 


.0509 


.4072 


.2036 


20 


859.46 


2.934226 


7.065774 


.0582 


.4654 


•2327 


30 


763 -97 


2.883076 


7. 116924 


•0655 


• 5236 


.2618 


40 


687.57 


2-837317 


7 162683 


.0727 


.5818 


.2909 


5 ° 


625.07 


2.795929 


7.204071 


.0800 


.6399 


.3200 


2 


572.99 


2.758147 


7-241853 


.0873 


.6981 


•349o 


10 


528.92 


2.723390 


7.276610 


•0945 


.7563 


.3781 


20 


491.14 


2.691205 


7-308795 


.1018 


.8144 


.4072 


30 


458 . 40 


2 661245 


7-338755 


. 1091 


.8726 


•4363 


40 


429.76 


2.633226 


7.366774 


.1164 


.9308 


•4654 


5 ° 


404 . 48 


2.606897 


7 393103 


1237 


.9889 


•4945 


3 


382.02 


2.582086 


7-4 I 79 I 4 


1309 


1.047 


•5235 


10 


361.91 


2.558601 


7.440399 


.1382 


1. 105 


.5526 


20 


343-82 


2-53633 1 


7.463669 


1454 


1. 163 


.5817 


30 


327.46 


2.515158 


7.484842 


1527 


1.222 


.6108 


40 


312.58 


2.494961 


7-505039 


1600 


1.280 


.6398 


. 5 ° 


298.99 


2.475657 


7-524343 


1673 


1-338 


.6689 


4 


286.54 


2.457185 


7-542815 


1746 


1.396 


.6980 


10 


275.08 


2-439459 


7 560541 


1818 


1-454 


•7^71 


20 


264 51 


2.422442 


7.577558 


1891 


1.51 2 


•756i 


30 


254-71 


2 . 406046 


7-593954 


1964 


i-57° 


.7852 


40 


245.62 


2.390264 


7.609736 


2036 


1 .629 


.8143 


* 5 ° 


237 16 


2.375041 


7.624959 


2109 


1.687 


•8433 


5 


229.26 


2.360328 


7.639672 


2182 


1-745 


.8726 


20 


214.94 


2.332317 


7.6676S3 


2328 


1. 861 


.9308 


40 


202.30 


2.305996 


7.694004 


2473 


1.977 


.9889 ! 


6 


191.07 


2.280193 


7.719307 


2619 


2.093 


1.047 


20 


181.03 


2-25775 1 


7.742249 


2764 


2.210 


1 -105 


40 


171.98 


2-235478 


7764522 


2910 


2.326 


1.163 


7 


163.80 


2.214314 


7-785686 


3055 


2.442 


1 .222 


20 


!56-37 


2-I94I53 


7.805847 


3201 


2.558 


1.280 


4 ° 


149.58 


2.174874 


7.825126 


3347 


2.674 


1-338 


8 


I43-36 


2. 156428 


7.843572 


3492 


2.790 


1.396 


20 


137-63 


2.138713 


7.861287 


363S 


2 906 


1-454 


40 


132.35 


2 121724 


7.878276 


3783 


3.022 


1.512 


9 


127-45 


2.105340 


7.894660 


3929 


3-I38 


1.570 


20 


122.91 


2.089587 


7.910413 


4075 


3- 2 54 


1.629 


«„ 4 ° 


118.68 


2.074378 


7.925622 


4220 


3-37° 


1.687 


10 


114.74 


2.059715 


7.940285 


4366 


3.486 


1-745 


.. 3 ° 


109.29 


2.038580 


7.961420 


4585 


3.660 


1.832 


11 c 


IQ4-33 


2.018409 


7-98i59i 


4803 


3 834 


1. 919 


*~ 3 ° 


99 8 £ 


1. 9991 74 


8.000826 


5022 


4.008 


2.006 


12 


95.67 


1 .980776 


8.019224 


5241 


4. 181 


2.093 


.« 3 ° 


91.86 


1 .963126 


8.036874 


5460 


4-355 


2. 181 


13 


88.34 


1. 946157 


8053843 


5679 


4.528 


2.268 


. 3 ° 


85.08 


1.929828 


8.070172 


5897 


4.701 


2-355 


14 


82.06 


191413 2 


8.085868 


6117 


4.875 


2.442 



TABLE XI. 



SQUARES, CUBES, ETC., OF NUMBERS 
FROM 1 TO 1042. 

Note.— If N be taken to represent any number in any 
column of this table, then the algebraic significance of the re- 
maining numbers, on the same line, in terms of N, will be as 
given in the following synopsis : 



N 


W 


w 


|/N 


fi 


1 

N 


|/N 


N 


i/W* 


fN 


fN 


1 

4/N 


ys 


fS§ 


N 


ys 


fN 


1 


N 2 


N 4 


N 6 


N 


fN 5 


1 

N 2 


N 3 


N 6 


N 9 


j/N 3 


N 


1 

N 3 


1 
N 


1 

N 2 


1 

N 3 


<3 


# 


N 

i 



TABLE 



SQUARES, CUBES, SQUARE AND CUBE ROOTS OP NUMBERS 



Mo. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


1 


1 


1 


10000000 


10000000 


•100000000 


2 


4 


8 


1-4142136 


1-2599210 


•500000000 


3 


9 


27 


1-7320508 


1-4422496 


•3333333?3 


4 


16 


64 


2-0000000 


1-5874011 


•250000000 


5 


25 


125 


2-2360680 


1-7099759 


•200000000 


6 


36 


216 


2-4494897 


1-8171206 


•166666667 


7 


49 


343 


26457513 


1-9129312 


•142857143 


8 


64 


512 


2-8284271 


2-0000000 


•125000000 


9 


81 


729 


3-0000000 


2-0800837 


•111111111 


10 


100 


1000 


3-1622777 


2-1544347 


•100000000 


11 


121 


1331 


3-3166248 


2-2239801 


•090909091 


12 


144 


1728 


3-4641016 


2-2894286 


•083333333 


13 


169 


2197 


3-6055513 


2-3513347 


•076923077 


14 


196 


2744 


3-7416574 


24101422 


•071428571 


15 


225 


3375 


38729833 


24662121 


•066666667 


16 


256 


4096 


40000000 


2-5198421 


•062500000 


17 


289 


4913 


4-1231056 


2-5712816 


•058823529 


18 


324 


5832 


4-2426407 


26207414 


•055555556 


19 


301 


6859 


4-3588989 


2-6684016 


•052631579 


20 


400 


8000 


4-4721360 


27144177 


•050000000 


21 


441 


9261 


45825757 


2-7589243 


•047619048 


22 


484 


10648 


4-6904158 


28020393 


•045454545 


23 


529 


12167 


4-7958315 


2-8438670 


•043478261 


24 


576 


13824 


4-8989795 


2-8844991 


•041666667 


25 


625 


15625 


5-0000000 


29240177 


•040000000 


26 


676 


17576 


5-0990195 


2-9624960 


038461538 


27 


729 


19683 


5-1961524 


3-0000000 


•037037037 


28 


784 


21952 


5-2915026 


30365889 


•035714286 


29 


841 


24389 


5-3851648 


30723168 


•034482759 


30 


900 


27000 


5-4772256 


31072325 


•033333333 


31 


961 


29791 


5-5677644 


31413806 


•032258065 


32 


1024 


32768 


5-6568542 


31748021 


•031250000 


33 


1089 


35937 


5-7445626 


32075343 


•030303030 


34 


1156 


39304 


5-8309519 


3-2396118 


•029411765 


35 


1225 


42875 


5-9160798 


32710663 


•028571429 


36 


1296 


46656 


6-0000000 


33019272 


•027777778 


37 


1369 


50653 


60827625 


3-3322218 


•027027027 


38 


1444 


54872 


61644140 


33619754 


•026315789 


39 


1521 


59319 


6-2449980 


33912114 


•025641026 


40 


1600 


64000 


6-3245553 


3-4199519 


•025000000 


41 


1681 


68921 


6-4031242 


3-4482172 


•024390244 


42 


1764 


74088 


6-4807407 


3-4760266 


•023809524 


43 


1849 


79507 


6-5574385 


3-5033981 


•023255814 


44 


1936 


85184 


6-6332496 


3-5303483 


•022727273 


45 


2025 


91125 


6-7082039 


3-5568933 


•022222222 


46 


2116 


97336 


6-7823300 


35830479 


•021739130 


47 


2209 


103823 


68556546 


3-6088261 


•021276600 


48 


2304 


110592 


6-9282032 


3-634241 1 


020833333 


49 


2401 


117649 


7-0000000 


3-6593057 


•020408163 


50 


2500 


125000 


7-0710678 


3-6840314 


•02C000000 



162 



SQUARES, CUBES, ETC., OF NUMBERS. 



163 



No. 


Squares. 


Cubes. 


Square Roots. 


, Cube Roots, 


Reciprocals. 


51 


2601 


132651 


7-1414284 


3-7084298 


•019607843 


52 


2704 


140608 


7-2111026 


3-7325111 


•019230769 


53 


2809 


148877 


7-2801099 


3-7562858 


•018867925 


54 


2916 


157464 


7-3484692 


3-7797631 


•018518519 


55 


3025 


166375 


7-4161985 


3-8029525 


•018181818 


56 


3136 


175616 


7 4833148 


3-8258624 


•017857143 


57 


3249 


185193 


7-5498344 


3-8485011 


•017543860 


58 


3364 


195112 


7-6157731 


3-8708766 


•017241379 


59 


3481 


205379 


7-6811457 


3-8929965 


•016949153 


60 


3600 


216000 


7-7459667 


3-9148676 


•016666667 


61 


3721 


226981 


7-8102497 


3-9364972 


•016393443 


62 


3844 


238328 


7-8740079 


3-9578915 


•016129032 


63 


3969 


250047 


7-9372539 


3-9790571 


•015873016 


64 


4096 


262144 


8-0000000 


4-0000000 


•015625000 


65 


4225 


274625 


8-0622577 


4-0207256 


•015384615 


66 


4356 


287496 


8-1240384 


40412401 


•015151515 


67 


4489 


300763 


8-1853528 


40615480 


•014925373 


68 


4624 


314432 


8-2462113 


4-0816551 


•014705882 


69 


4761 


328509 


83066239 


41015661 


•014492754 


70 


4900 


343000 


8-3666003 


41212853 


•014285714 


7J 


5041 


357911 


8-4261498 


4-1408178 


•014084517 


72 


5184 


373248 


8-4852814 


4-1601676 


•013888889 


73 


5329 


389017 


8-5440037 


4-1793390 


•013698630 


74 


5476 


405224 


8-6023253 


4-1983364 


•013513514 


75 


5625 


421875 


86602540 


4-2171633 


•013333333 


76 


5776 


438976 


8-7177979 


4-2358236 


•013157895 


77 


5929 


456533 


8-7749644 


4-2543210 


•012987013 


78 


6084 


474552 


8-8317609 


4-2726586 


•012820513 


79 


6241 


493039 


8-8881944 


4-2908404 


•012658228 


80 


6400 


512000 


8-9442719 


4-3088695 


•012500000 


81 


6561 


531441 


90000000 


4-3267487 


•012345679 


82 


6724 


551368 


90553851 


43444815 


012195122 


83 


6889 


571787 


91104336 


4-3620707 


012048193 


84 


7056 


592704 


9-1651514 


4-3795191 


•011904762 


85 


7225 


614125 


9-2195445 


4-3968296 


011764706 


86 


7396 


636056 


9-2736185 


4-4140049 


•011627907 


87 


7569 


658503 


9-3273791 


4-4310476 


01 1494253 


88 


7744 


681472 


9-3808315 


4-4479602 


•011363636 


89 


7921 


704969 


9-4339811 


4-4647451 


•011235955 


90 


8100 


729000 


94868330 


4-4814047 


011111111 


91 


8281 


753571 


9-5303920 


4-4979414 


•010989011 


92 


8464 


778688 


9-5916630 


45143574 


•010869565 


93 


8649 


804357 


9-6436508 


4-5306549 


010752688 


94 


8836 


830584 


9-6953597 


4-5468359 


•010638298 


95 


9025 


857375 


9-7467943 


4-5629026 


•010526316 


96 


9216 


884736 


9-7979590 


4-5788570 


•010416667 


97 


9409 


912673 


9-8488578 


45947009 


•010309278 


98 


9604 


941192 


9-8994949 


4-6104363 


•010204082 


99 


9801 


970299 


9-9498744 


4-6260650 


010101010 


100 


10000 


1000000 


100000000 


4-6415888 


010000000 


101 


10201 


1030301 


100498756 


4-6570095 


•009900990 


102 


1U404 


1061208 


100995049 


4-6723287 


•009803922 


303 


10609 


1092727 


101488916 


4-6875482 


•009708738 


104 


10816 


1124864 


101980390 


4-7026694 


009615385 


105 


11025 


1157625 


10*2469508 


4-7176940 


•009523810 


J06 


11236 


1191016 


10-2956301 


4-7326235 


•009433962 


107 


11449 


1225043 


10-3440804 


4-7474594 


•009345794 


108 


11664 


1259712 


10-3923048 


4-7622032 


•009259259 


109 


11881 


1295029 


10-4403065 


4-7768562 


009174312 


110 


12100 


1331000 


10-4880885 


4-7914199 


•009090909 


111 


12321 


1367631 


10-5356538 


4-8058995 


•00900900S 


112 


12544 


1404928 


10-5830052 


4-8202845 


•008928571 



164 



SQUARES, CUBES, ETC., OF NUMBERS. 



No. 


Squares. 


Cubes. 


Square Roots 


Cube Roots. 


Reciprocals. 


113 


12769 


1442897 


10-6301458 


4-8345881 


•008849558 


]J4 


129.96 


1481544 


10-6770783 


4-8488076 


•008771930 


115 


13225 


1520875 


107238053 


4-8629442 


•008695652 


110 


13456 


1560896 


10-7703296 


4-8769990 


•008620690 


117 


13689 


1601613 


10 8166538 


4-8909732 


•008547009 


118 


13924 


1643032 


10-8627805 


4-9048681 


•008474576 


119 


14161 


1685159 


10 9087121 


4-9186847 


•008403361 


120 


14400 


1728000 


109544512 


4-9324242 


•008333333 


121 


14641 


1771561 


n-ooooooo 


4-9460874 


•008264463 


122 


14884 


18.15848 


110453610 


4-9596757 


•008196721 


123 


15129 


1860867 


11 0905365 


4-9731898 


•008130081 


124 


15376 


1906624 


11 1355287 


4-9866310 


•008064516 


125 


15625 


• 1953125 


1 1-1803309 


5-0000000 


•008000000 


126 


15876 


2000376 


11-2249722 


50132979 


•007936508 


127 


16129 


2048383 


11-2694277 


5-0265257 


•007874016 


128 


16384 


2097152 


11-3137085 


50396842 


•007812500 


129 


16641 


2146689 


11-3578167 


50527743 


•007751938 


130 


16900 


2197000 


U-4017543 


50657970 


•007692308 


131 


17161 


2248091 


11-4455231 


50787531 


007633588 


132 


17424 


2299968 


11-4891253 


5-0916434 


•007575758 


133 


17689 


2352637 


11-5325626 


51044687 


•007518797 


134 


17956 


2406104 


11-5758369 


5- 1172299 


•007462687 


135 


18225 


2460375 


11-6189500 


5- 1299278 


•007407407 


136 


18496 


2515456 


11-6619038 


51425632 


•007352941 


137 


18769 


2571353 


11-7046999 


5-1551367 


•007299270 


138 


19044 


2628072 


11-7473401 


51676493 


•007246377 


139 


19321 


2685619 


11-7898261 


51801015 


007194245 


140 


19600 


2744000 


11-8321596 


51 924941 


•007142857 


141 


19881 


2803221 


11-8743421 


5-2048279 


•007092199 


142 


20164 


2863288 


11-9163753 


5-2171034 


•007042254 


143 


20449 


2924207 


11 '9582607 


5-2293215 


•006993007 


144 


20736 


2985984 


12-0000000 


5-2414828 


006944444 


145 


21025 


3048625 


120415946 


5-2535879 


•006896552 


146 


21316 


3112136 


120830460 


5-2656374 


•006849315 


147 


21609 


3176523 


121243557 


5-2776321 


•006802721 


148 


21904 


3241792 


121655251 


5-2895725 


•006756757 


149 


22201 


3307949 


12-2065556 


5 3014592 


•006711409 


150 


22500 


3375000 


12-2474487 


5-3132928 


•006666667 


151 


22801 


3442951 


12*2882057 


5-3250740 


•006622517 


152 


23104 


3511808 


12-3288280 


5-3368033 


•006578947 


153 


23409 


3581577 


12-3693169 


5-3484812 


•006535948 


154 


23716 


3652264 


12-4096736 


5-3601084 


•006493506 


155 


24025 


3723875 


12-4498996 


5-3716854 


•006451613 


156 


24336 


3796416 


12-4899960 


5-3832126 


•006410256 


157 


24649 


3869893 


12-5299641 


5-3946907 


•006369427 


158 


24964 


3944312 


125698051 


5-4061202 


•006329114 


159 


25281 


4019679 


12-6095202 


5-4175015 


•006289308 


160 


25600 


4096000 


12-6491108 


5-4288352 


•006250000 


161 


25921 


4173281 


12 6885775 


5-4401218 


•006211180 


162 


26244 


4251528 


12-7279221 


5-4513618 


•006172840 


163 


26569 


4330747 


12-7671453 


54625556 


•006134969 


164 


26896 


4410944 


12-8062485 


5-4737037 


•006097561 


165 


27225 


4492125 


12-8452326 


5-4848066 


•006060606 


166 


27556 


4574296 


128840987 


54958647 


•006024096 


167 


27889 


4657463 


12-9228480 


5-5068784 


•005988024 


168 


28224 


4741632 


129614814 


5-5178484 


•005952381 


169 


28561 


4826809 


130000000 


5-5287748 


•005917160 


170 


28900 


4913000 


130384048 


5-5396583 


•005882353 


171 


29241 


5000211 


13 0766968 


5-5504991 


•005847953 


172 


29584 


5088448 


13 1148770 


5-5612978 


•005813953 


173 


29929 


5177717 


13-1529464 


5-5720546 


•005780347 


174 


30276 


5268024 


13-1909060 


5-5827702 


005747126 



SQUARES, CUBES, ETC., OF NUMBERS. 



165 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


175 


30625 


5359375 


132287566 


55934447 


005714286 


176 


30976 


5451776 


13-2664902 


5-6040787 


•005681818 


177 


31329 


5545233 


133041347 


5 6146724 


•005649718 


178 


31684 


5639752 


13 3416641 


56252263 


•005617978 


179 


32041 


5735339 


133790882 


5 6357408 


•005586592 


180 


32400 


5832000 


13 4164079 


56462162 


•005555556 


181 


32761 


5929741 


134536240 


5 6566528 


•005524862 


182 


33124 


6028568 


134907376 


5 6670511 


•005494505 


183 


33489 


6128487 


13 5277493 


5-6774114 


•005404481 


184 


33856 


6229504 


13-5646600 


56877340 


•005434783 


185 


34225 


6331625 


136014705 


5-6980192 


•005405405 


186 


34596 


6434856 


13-6381817 


5-7082675 


•005376344 


187 


34969 


6539203 


136747943 


5-7184791 


•005347594 


188 


35344 


6644672 


137113092 


5-7286543 


•005319149 


189 


35721 


6751269 


137477271 


5-7387936 


•005291005 


190 


36100 


6859000 


13-7840488 


5-7488971 


■005263158 


191 


36481 


6967871 


13-8202750 


5-7589652 


•005235602 


192 


36864 


7077888 


138564065 


5-7 '>89982 


•005208333 


193 


37249 


7189017 


13-8924440 


5-7789966 


•005181347 


194 


37636 


7301384 


13-9283883 


5-7889604 


•005154639 


195 


38025 


7414875 


139642400 


5-7988900 


•005128205 


196 


38416 


7529536 


140000000 


5-8087857 


•005102041 


197 


38809 


7645373 


140356688 


5-8186479 


•005076142 


198 


39204 


7762392 


140712473 


58284767 


•005050505 


199 


39601 


7880599 


141067360 


58382725 


•005025126 


200 


40000 


8000000 


14- 1421356 


5 8480355 


•005000000 


201 


40401 


8120601 


14- 1774469 


5-8577660 


•001975124 


202 


40804 


8242408 


14*2126704 


5-8074643 


•004950495 


203 


41209 


836542', 


14-2478068 


5-8771307 


•004926108 


204 


41616 


8489664 


14 2828569 


58867653 


•004901961 


205 


42025 


8615125 


14*317821 1 


5-8963685 


•004878049 


206 


42436 


8741816 


14-3527001 


59059406 


•004854369 


207 


42*49 


8869743 


143874946 


5-9154817 


•004830918 


208 


43264 


8998912 


14*4222051 


5-9249921 


•004807692 


209 


43681 


9129329 


14-4568323 


5-9344721 


•004784689 


210 


44100 


9261000 


14-4913767 


59439220 


•004761905 


211 


44521 


9393931 


14-5258390 


5-9533418 


•004739336 


212 


44944 


9528128 


145602198 


5-9027320 


•004716981 


213 


45369 


9663597 


145945195 


5-9720926 


•004094836 


214 


45796 


9800344 


14-6287388 


5-9814240 


•004072897 


215 


46225 


9938375 


14-6628783 


5-9907264 


•004651163 


216 


46656 


10077696 


14 6969385 


60000000 


•004629630 


217 


47089 


10218313 


147309199 


60092450 


•004608295 


218 


47524 


10360232 


147648231 


6-0184617 


•004587156 


219 


47961 


10503459 


14-7986486 


60276502 


•004566210 


220 


48400 


10648000 


148323970 


60368107 


•004545455 


221 


48841 


10793861 


148660687 


6 0459435 


•004524887 


222 


49284 


10941048 


148996644 


60550489 


•004504505 


223 


49729 


11089567 


149331845 


60641270 


•004484305 


224 


50176 


11239424 


14*9666295 


60731779 


•004464286 


225 


506-25 


11390625 


15-OuOOOOO 


60822020 


•004444444 


226 


51076 


11543176 


15-0332964 


6-0911994 


•004424779 


227 


51529 


11097083 


150665192 


61001702 


•004405286 


228 


51984 


1 1852352 


15 0996689 


6-1091147 


•004385965 


229 


52441 


12008989 


151327460 


6-1180332 


•004366812 


230 


52900 


12167000 


15- 1657509 


6-1269257 


•004347826 


231 


53361 


12326391 


151986842 


6-1357924 


•004329004 


232 


53824 


12487168 


15-2315462 


61446337 


•004310345 


233 


54289 


12649337 


152643375 


6*1534495 


•004291845 


234 


54756 


12812904 


15-2970585 


6-1622401 


•004273504 


235 


55225 


12977875 


15-3297097 


61710058 


•004255319 


236 


55696 


13144256 


153622915 


61797466 


•004237288 



166 



SQUARES, CUBES, ETC., OF NUMBERS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocal*. 


237 


56169 


13312053 


153948043 


6-1884628 


•004219409 


238 


56644 


13481272 


154272486 


61971544 


•004201681 


239 


57121 


13651919 


15-4596248 


6-2058218 


•004184100 


240 


57600 


13824000 


15-4919334 


6-2144650 


•004166667 


241 


58081 


13997521 


15-5241747 


6-2230843 


•004149378 


242 


58564 


14172488 


15-5563492 


6-2316797 


•004132231 


243 


59049 


14348907 


15-5884573 


6-2402515 


•004115226 


244 


59536 


14526784 


15-6204994 


6-2487998 


•004098361 


245 


60025 


14706125 


15-6524758 


6-2573248 


•004081633 


246 


60516 


14886936 


15-6843871 


6-2658266 


•004065041 


247 


61009 


15069223 


157162336 


6-2743054 


•004048583 


248 


61504 


15252992 


15-7480157 


6-2827613 


•004032258 


249 


62001 


15438249 


15-7797338 


6-2911946 


•004016064 


250 


62500 


15625000 


15-8113883 


6-2996053 


•004000000 


251 


63001 


15813251 


15-8429795 


6-3079935 


•003984064 


252 


63504 


16003008 


15-8745079 


6-3163596 


•003968254 


253 


64009 


16194277 


15-9059737 


6-3247035 


•003952569 


254 


64516 


16387064 


15-9373775 


6-3330256 


•003937008 


255 


65025 


16581375 


15-9687194 


6-3413257 


•003921569 


256 


65536 


16777216 


160000000 


6-3496042 


•003906250 


257 


66049 


16974593 


160312195 


6-3578611 


•003891051 


258 


66564 


17173512 


160623784 


6-3660968 


•003875969 


259 


67081 


17373979 


160934769 


6-3743111 


•003861004 


260 


67600 


17576000 


161245155 


6-3825043 


•003846154 


261 


68121 


17779581 


16-1554944 


6-3906765 


•003831418 


262 


68644 


17984728 


161864141 


6*3988279 


•003816794 


263 


69169 


18191447 


16 2172747 


6-4069585 


•003802281 


264 


69696 


18399744 


16-2480768 


6-4150687 


•003787879 


265 


70225 


18609625 


16-2788206 


6-4231583 


•003773585 


266 


70756 


18821096 


163095064 


6-4312276 


•003759398 


267 


71289 


19034163 


16-3401346 


6-4392767 


•003745318 


268. 


71824 


19248832 


16-3707055 


6-4473057 


•003731343 


269 


72361 


19465109 


16-4012195 


6-4553148 


•003717472 


270 


72900 


19683000 


16-4316767 


6-4633041 


•003703704 


271 


73441 


19902511 


16-4620776 


6-4712736 


•003690037 


272 


73984 


20123648 


16-4924225 


6-4792236 


•003676471 


273 


74529 


20346417 


16-5227116 


6-4871541 


•003663004 


274 


75076 


20570824 


16-5529454 


6-4950653 


•003649635 


275 


75625 


20796875 


16-5831240 


65029572 


•003636364 


276 


76176 


21024576 


16-6132477 


6-5108300 


•003623188 


277 


76729 


21253933 


16-6433170 


6-5186839 


•003610108 


278 


77284 


21484952 


16-673332") 


6-5265189 


•003597122 


279 


77841 


21717639 


16-7032931 


6-5343351 


•003584229 


280 


78400 


21952000 


16-7332005 


65421326 


•003571429 


281 


78961 


22188041 


16-7630546 


6-5499116 


•003.558719 


282 


79524 


22425768 


16-7928556 


65576722 


•003546099 


283 


80089 


22665187 


16-8226038 


65654144 


•003533569 


284 


80656 


22906304 


16-8522995 


65731385 


•003522127 


285 


81225 


23149125 


168819430 


6-5808443 


•003508772 


286 


81796 


23393656 


16-9115345 


65885323 


•003496503 


287 


82369 


23639903 


16-9410743 


65962023 


•003484321 


288 


82944 


23887872 


16-9705627 


66038545 


•003472222 


289 


83521 


24137569 


17-0000000 


66114890 


•003460208 


290 


84100 


24389000 


17-0293864 


66191060 


003448276 


291 


84681 


24642171 


170587221 


6-6267054 


•003436426 


292 


85264 


24897088 


17-0880075 


66342874 


•003424(558 


293 


85849 


25153757 


171172428 


0-6418522 


■003412969 


294 


86436 


25412184 


171464282 


6-6493998 


•003401361 


295 


87025 


25672375 


171755640 


66569302 


•003389831 


296 


87616 


25934336 


17-2046505 


6-6644437 


•003378378 


297 


88209 


26198073 


17-2336879 


6 6719403 


•003367003 


298 


88804 


26463592 


17-2626765 


6-6794200 


■003355705 



SQUARES, CUBE J, ETC., OF NUMBERS. 



167 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


299 


89401 


26730899 


17-2916165 


6-6868831 


•003344482 


300 


90000 


27000000 


17-3205081 


6-6943295 


•003333333 


301 


90601 


27270901 


17-3493516 


6-7017593 


•003322259 


302 


91204 


27543608 


17-3781472 


6-7091729 


•003311258 


303 


91809 


27818127 


17-4068952 


6-7165700 


•003301330 


304 


92416 


28094464 


17-4355958 


6-7239508 


•003289474 


305 


93025 


28372625 


17-4642492 


6-7313155 


•003278689 


306 


93636 


28652616 


17-4928557 


6-7386641 


•003267974 


307 


94249 


28934443 


17-5214155 


6-7459967 


•003257329 


308 


94864 


29218112 


17-5499288 


6-7533134 


•003246753 


309 


95481 


29503629 


17-5783958 


6-7606143 


•003236246 


310 


96100 


29791000 


17-6068169 


6-7678995 


•003225806 


311 


96721 


30080231 


17-6351921 


6-7751690 


•003215434 


312 


97344 


30371328 


17-6635217 


6-7824229 


•003205128 


313 


97969 


30664297 


17-6918060 


6-7896613 


•003194888 


314 


98596 


30959144 


17-7200451 


6 7968844 


•003184713 


315 


99225 


31255875 


17-7482393 


6-8040921 


•003174603 


316 


99856 


31554496 


17-7763883 


6-8112847 


•003164557 


317 


100489 


31855013 


17-8044938 


6-8184620 


•003154574 


318 


101124 


32157432 


17-8325545 


6-8250242 


•003144654 


319 


101761 


32461759 


17-8605711 


6-8327714 


•003134796 


320 


102400 


32768000 


17-8885438 


6-8399037 


•003125000 


321 


103041 


33076161 


17-9164729 


6-8470213 


•003115265 


322 


103684 


33386248 


17-9443584 


6-8541240 


•003105590 


323 


104329 


33698267 


17-9722008 


6-8612120 


•003095975 


324 


104976 


34012224 


18-0000000 


6-8682855 


•003086420 


325 


105625 


34328125 


18-0277564 


687534 13 


•003076923 


326 


106276 


34645976 


18-0554701 


6-8823888 


003067485 


327 


106929 


34965783 


18-0831413 


6-8894188 


•003058104 


328 


107584 


35287552 


18-1107703 


6-8964345 


•003048780 


329 


-108241 


35611289 


18-1383571 


6-9034359 


•003039514 


330 


108900 


35937000 


18-1659021 


69104232 


•003030303 


331 


109561 


36264691 


18-1934054 


6-9173964 


•003021148 


3.32 


110224 


36594368 


18-2208672 


6-9243556 


•003012048 


333 


110889 


36926037 


18-2482876 


6-9313008 


•003003003 


334 


111556 


37259704 


18-2756669 


6-9382321 


•002994012 


335 


112225 


37595375 


18-3030052 


6-9451496 


•002985075 


335 


112896 


37933056 


18-3303028 


6-9520533 


•002976190 


337 


113569 


38272753 


18-3575598 


6-9589434 


•002967359 


338 


114244 


38614472 


18-3847763 


6-9658198 


•002958580 


339 


114921 


38958219 


18-4119526 


6-9726826 


•002949853 


340 


115600 


39304000 


18-4390889 


6-9795321 


•002941176 


341 


116281 


39651821 


18 4661853 


6-9863681 


•002932551 


342 


116964 


40001688 


18-4932420 


6-9931906 


•002923977 


343 


117649 


40353607 


18-5202592 


7-0000000 


•002915452 


344 


118336 


40707584 


18-5472370 


7-0067962 


•002906977 


345 


119025 


41063625 


18-5741756 


70135791 


•002898551 


346 


119716 


41421736 


18-6010752 


70203490 


•002890173 


347 


120409 


41781923 


18-6279360 


7-0271058 


•002881844 


348 


121104 


42144192 


18-6547581 


7-0338497 


•002873563 


343 


121801 


42508549 


18-6815417 


7-0405806 


•002865330 


350 


122500 


42875000 


18-7082869 


7-0472987 


•002857143 


351 


123201 


43243551 


18-7349940 


7-0540041 


•002849003 


352 


123904 


43614208 


18-7616630 


7-0606967 


•002840909 


353 


124609 


43986977 


18-7882942 


7-0673767 


•002832861 


354 


125316 


44361864 


18-8148877 


7-0740440 


•002824859 


355 


126025 


44738875 


188414437 


7-0806988 


•002816901 


356 


126736 


45118016 


188679623 


70873411 


•002808989 


357 


127449 


45499293 


188944436 


7-0939709 


•002801120 


358 


128164 


45862712 


18-9208879 


7-1005885 


•002793296 


359 


128881 


46268279 


189472953 


71071937 


•002785515 


360 J 


129600 


46656000 


18-9736660 1 


71137866 


•002777778 



168 



SQUARES, CUBES, ETC., OF NUMBERS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


361 


130321 


47045881 


190000000 


71203674 


002770083 


362 


131044 


47437928 


190262976 


71269360 


•002762431 


363 


131769 


47832147 


190525589 


71334925 


•002754821 


364 


132496 


48228544 


190787840 


71400370 


•002747253 


365 


133225 


48627125 - 


191049732 


71465695 


•002739726 


366 


133956 


49027896 


191311265 


71530901 


•002732240 


367 


134689 


49430863 


191572441 


71595988 


•002724796 


368 


135424 


49836032 


191833261 


71660957 


•002717391 


369 


136161 


50243409 


192093727 


7 1725809 


•002710027 


370 


136900 


50653000 


192353841 


7 1790544 


•002702703 


371 


137641 


51064811 


192613603 


71855162 


•002695418 


372 


138384 


51478848 


192873015 


71919663 


•002688172 


373 


139129 


51895117 


193132079 


71984050 


•002680965 


374 


139876 


52313624 


193390796 


7 2048322 


002673797 


375 


140625 


52734375 


193649167 


7-2112479 


•002666667 


376 


141376 


53157376 


193907194 


7 2176522 


002659574 


377 


142129 


53582633 


19 4164878 


7-2240450 


•002652520 


378 


142884 


54010152 


194422221 


7-2304268 


•002645503 


379 


143641 


54439939 


194679223 


72367972 


•002638521 


380 


144400 


54872000 


194935887 


7 2431565 


•002631579 


381 


145161 


55306341 


195192213 


72495045 


•002624672 


382 


145924 


55742968 


195448203 


7 2558415 


002617801 


383 


146689 


56181887 


195703858 


72621675 


002610966 


384 


147456 


56623104 


195959179 


7-2684824 


002604167 


385 


148225 


57006625 


196214169 


7-2747864 


•002597403 


386 


148996 


57512456 


19 6468827 


7-2810794 


•002590674 


387 


149769 


57960603 


196723156 


7-2873617 


•002583979 


388 


150544 


58411072 


19 6977156 


7-2936330 


•002577320 


389 


151321 


58863869 


197230829 


7-2998936 


•002570694 


390 


152100 


59319000 


19 7484177 


7-3061436 


•002564103 


391 


152881 


59776471 


197737199 


7-3123828 


002557545 


392 


153664 


60230288 


197989899 


7-3186114 


002551020 


393 


154449 


60698457 


198242276 


7-3248295 


•002544529 


394 


155236 


61162984 


198494332 


73310369 


•002538071 


395 


156025 


61629875 


19 8746069 


7-3372339 


•002531646 


396 


156816 


62099136 


198997487 


73434205 


•002525253 


397 


157609 


62570773 


19-9248588 


7-3495966 


•002518892 


398 


158404 


63044792 


199499373 


7-3557624 


•002512563 


o99 


159201 


63521199 


199749844 


7-3619178 


•002506266 


400 


160000 


64000000 


200000000 


7-3680630 


•002500000 


401 


160801 


64481201 


20 0249844 


7-3741979 


•002493766 


402 


161604 


64964808 


200499377 


7-3803227 


•002487562 


403 


162409 


65450827 


20 0748599 


7-3864373 


•002481390 


404 


163216 


65939264 


20 0997512 


73925418 


•002475248 


405 


164025 


66430125 


201246118 


7-3986363 


002469136 


406 


164836 


66923416 


201494417 


7-4047206 


002463054 


407 


165649 


67419143 


201742410 


7 4107950 


002457002 


408 


166464 


67917312 


201990099 


7-4168595 


•002450980 


409 


167281 


68417929 


20 2237484 


74229142 


002444988 


410 


168100 


68921000 


20 2484567 


7-4289589 


002439024 


411 


168921 


69426531 


20 2731349 


74349938 


•002433090 


412 


169744 


69934528 


20-2977831 


74410189 


002427184 


413 


170569 


70444997 


20 3224014 


7 4470342 


002421308 


414 


171396 


70957944 


20 3469899 


74530399 


002415459 


415 


172225 


71473375 


20-3715488 


7 4590359 


002409639 


416 


173056 


71991296 


20-3960781 


7-4650223 


002403846 


417 


173889 


72511713 


20-4205779 


7-4709991 


002398082 


418 


174724 


73034632 


20-4450483 


7-4769064 


•002392344 


419 


175561 


73560059 


20-4694895 


74829242 


•002386635 


420 


176400 


74088000 


20-4939015 


7-4888724 


002380952 


421 


177241 


74618461 


20-5182845 


7-4948113 


002375297 


422 


178084 


75151448 


20-5426386 


7-5007406 


•002369668 



SQUARES, CUBES, ETC., OF NU3fBERS. 



169 



Nw. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


423 


178929 


75686967 


20 5669638 


7-5066607 


•002364066 


424 


179776 


7622.5024 


20-5912603 


7-5125715 


002358491 


425 


180625 


76765625 


20 6155281 


7-5184730 


002352941 


426 


181476 


77308776 


206397674 


7-5243652 


002347418 


427 


182329 


77854483 


20-6639783 


7-5302482 


•002341920 


428 


18?184 


78402752 


20-6881609 


75361221 


•002336449 


429 


184041 


78953589 


20-7123152 


7-5419867 


002331002 


430 


184900 


79507000 


207364414 


7-5478423 


002325581 


431 


185761 


80062991 


207605395 


7-5536888 


002320186 


432 


186624 


80621568 


20-7846097 


7-5595263 


002314815 


433 


187489 


81182737 


208086520 


7-5653548 


002309469 


434 


188356 


81746504 


20-8326667 


7-5711743 


002304147 


435 


189225 


82312875 


20-8566536 


7-5769849 


002298851 


430 


190096 


82881856 


208806130 


7-5827865 


002293578 


437 


190909 


83453453 


20 9045450 


7-5885793 


002288330 


438 


191844 


84027672 


20-9284495 


7-5943633 


002283105 


439 


192721 


84604519 


209523268 


7-6001385 


002277904 


440 


193600 


85184000 


209761770 


76059049 


•002272727 


441 


194481 


85766121 


210000000 


7-6116626 


•002267574 


442 


195364 


86350888 


210237960 


7-6174 116 


•002262443 


443 


196249 


86938307 


210475652 


7-6231519 


•002257336 


444 


197136 


87528384 


210713075 


7-6288837 


002252252 


445 


198025 


88121125 


210950231 


7-6346067 


002247191 


446 


198916 


88716536 


211187121 


7-6403213 


002242152 


447 


199809 


80314623 


211423745 


7-6460272 


•002237136 


448 


200704 


89915392 


21-1660105 


7-6517247 


•002232143 


448 


200704 


90518849 


21-1896201 


7-6574138 


002227171 


449 


201601 


91125000 


21 2132034 


7-6630943 


002222222 


450 


. 202500 


91733851 


212367606 


7-6687665 


002217295 


452 


204304 


92345408 


21-2602916 


76744303 


002212389 


453 


205209 


92959677 


21-2837967 


7-6800857 


•002207506 


454 


206116 


93576664 


21-3072758 


7-6857328 


002202643 


455 


207025 


94196375 


21-3307290 


7 6913717 


•002197802 


456 


207936 


94818816 


21-3541565 


76970023 


•002192982 


457 


208849 


95443993 


21-3775583 


7 7026246 


•002188184 


458 


209764 


96071912 


21-4009346 


7-7082388 


•002183406 


459 


210681 


96702579 


214242853 


7-7188448 


002178649 


400 


211600 


97336000 


214476106 


7-7194426 


•002173913 


461 


212521 


97972181 


21-4709106 


7 7250325 


•002169197 


402 


213444 


98611128 


21-4941853 


7 7306141 


002164502 


403 


214369 


99252847 


21-5174348 


7-7361877 


•002159827 


464 


215296 


99897344 


21-5406592 


7-7417532 


002155172 


405 


216225 


- 100544625 


21-5G38587 


7-7473109 


•002150538 


406 


217156 


101194696 


21-5870331 


7-7528606 


•002145923 


467 


218089 


101847563 


21-6101828 


7-7584023 


•002141328 


468 


219024 


102503232 


216333077 


7-7639361 


002136752 


469 


219961 


103161709 


21-6564078 


7-7694620 


002132196 


470 


22J900 


103823000 


21-6794834 


7-7749801 


002127660 


471 


221841 


104487111 


21-7025344 


7-7804904 


002123142 


472 


222784 


105154048 


21-7255610 


7-7859928 


002118644 


473 


223729 


105823817 


217485632 


77914875 


002114165 


474 


224676 


106496424 


217715411 


7-7969745 


002109705 


475 


225625 


107171875 


217944947 


7-8024538 


002105263 


476 


226576 


107850176 


218174242 


78079254 


002100840 


477 


• 227529 


108531333 


218403297 


7-8133892 


002096486 


478 


228484 


109215352 


218632111 


7-8188456 


002092050 


479 


229441 


100902239 


21-8860686 


7-8242942 


002087683 


480 


230400 


110592000 


219089023 


78297353 


•002083333 


48] 


231361 


111284641 


219317122 


7-8351688 


002079002 


482 


232324 


111980168 


21-9544984 


7 8405949 


002074689 


483 


233289 


112678587 


219772610 


78460134 


•002070393 


484 


234256 


113379904 


220000000 


7 8514244 


•002066116 



170 



SQUARES, CUBES, ETC., OF NUMBERS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


485 


235225 


114084125 


1 220227155 


7-8568281 


002061856 


486 


236196 


114791256 


220454077 


78622242 


002057613 


487 


237169 


115501303 


220680765 


78676130 


•002053388 


488 


238144 


116214272 


22O907220 


7-8729944 


002049180 


489 


239121 


116930169 


22 1133444 


7-8783684 


002044990 


490 


240100 


117649000 


221359436 


78837352 


002040816 


491 


241081 


118370771 


221585198 


7-8890946 


002036660 


492 


242064 


119095488 


221810730 


7-8944468 


002032520 


493 


243049 


119823157 


222036033 


7-8997917 


•002028398 


494 


244036 


120553784 


22 2261108 


7-9051294 


002024291 


495 


245025 


121287375 


22 2485955 


79104599 


•002020202 


496 


246016 


122023936 


222710575 


79157832 


002016129 


497 


247009 


122763473 


22-2934968 


79210994 


002012072 


498 


248004 


123505992 


223159136 


79264085 


•002008032 


499 


249001 


124251499 


22 3383079 


79317104 


002004008 


500 


250000 


125000000 


22-3606798 


79370053 


•002000000 


501 


251001 


125751501 


22 3830293 


7 9422931 


•001996008 


502 


252004 


126506008 


22-4053565 


7-9475739 


•001992032 


503 


253009 


127263527 


22-4276615 


79528477 


•001988072 


504 


254016 


128024064 


224499443 


7-9581144 


001984127 


505 


255025 


128787625 


22 4722051 


7-9633743 


•001980198 


506 


256036 


129554216 


22 4944438 


7-9686271 


001976285 


507 


257049 


130323843 


22 5166605 


79738731 


001972387 


508 


258064 


131096512 


22 5388553 


7-9791122 


•001968504 


509 


259081 


131872229 


22 5610283 


79843444 


•001964637 


510 


260100 


132651000 


22-5831796 


79895697 


•001960785 


511 


261121 


133432831 


22-6053091 


7-9947883 


001956947 


512 


262144 


134217728 


22-6274170 


80000000 


•001953125 


513 


263169 


135005697 


226495033 


80052049 


001949318 


514 


264196 


335796744 


22-6715681 


80104032 


•001945525 


515 


265225 


136590875 


22 6936114 


80155946 


•001941748 


516 


266256 


137388096 


227156334 


80207794 


•001937984 


517 


267289 


138188413 


227376341 


8 0259574 


•001934236 


518 


268324 


138991832 


22 7596134 


8 0311287 


•001930502 


519 


269361 


139798359 


22-7815715 


80362935 


•001926782 


520 


270400 


140608000 


22 8035085 


80414515 


•001923077 


521 


271411 


141420761 


22 8254244 


80466030 


•001919386 


522 


272484 


142236648 


22 8473193 


80517479 


•001915709 


523 


273529 


143055667 


228691933 


8-0568862 


001912046 


524 


274576 


143877824 


228910463 


80620180 


•001908397 


525 


275625 


144703125 


22-9128785 


80671432 


001904762 


526 


276676 


145531576 


22 934G899 


80722620 


•001901141 


527 


277729 


146363183 


22 9564806 


8 0773743 


•001897533 


528 


278784 


147197952 


22-9782506 


80824800 


•001893939 


529 


279841 


148035889 


230000000 


80875794 


001890359 


530 


280900 


148877001 


23 0217289 


80926723 


•001886792 


531 


281961 


149721291 


23 0434372 


80977589 


001883239 


532 


283024 


150568768 


230651252 


81028390 


•001879699 


533 


284089 


151419437 


23 0867928 


81079128 


•001876173 


534 


285156 


152273304 


231084400 


81129803 


•001872659 


535 


286225 


153130375 


231300670 


81180414 


:001869159 


536 


287296 


153990656 


231516738 


8-1230962 


•001865672 


537 


288369 


154854153 


231732605 


81281447 


•001862197 


538 


289444 


155720872 


231948270 


81331870 


•001858736 


539 


290521 


156590819 


23 2163735 


8-1382230 


001855288 


540 


291600 


157464000 


23 2379001 


81432529 


001851852 


541 


292681 


158340421 


23 2594067 


81482705 


•001848429 


542 


293764 


159220088 


23 2808935 


8-1532939 


001845018 


543 


294849 


160103007 


23 3023604 


81583051 


•001841621 


544 


295936 


160389184 


23-3238076 


81633102 


•001838235 


545 


297025 


161878625 


23 3452351 


8-1683092 


•001834S0-J 


546 


298116 


162771336 


23 3666429 


81733020 


•001831502 



SQUARES, CUBES, ETC., OF NUMBERS. 



171 



No. 


Scuares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


547 


299209 


163667323 


23-3880311 


8-1782888 


•001828154 


548 


300304 


164566592 


23-4093998 


81832695 


•001824818 


549 


301401 


165469149 


23-4307490 


8-1882441 


•001821494 


550 


302500 


166375000 


23-4520788 


8-1932127 


•001818182 


551 


303601 


167284151 


23-4733892 


8-1981753 


•001814882 


552 


304704 


168196608 


23-4946802 


8-2031319 


•001811594 


553 


305809 


169112377 


235159520 


8-2080825 


•001808318 


554 


306916 


170031464 


23-5372046 


8-2130271 


•001805054 


555 


308025 


170953875 


23-5584380 


8-2179657 


•001801802 


556 


309136 


171879616 


23-5796522 


8-2228985 


•001798561 


557 


310249 


172808693 


23-6008474 


8-2278254 


•001795332 


558 


311364 


173741112 


23-6220236 


8-2327463 


•001792115 


559 


312481 


174676879 


23-6431808 


82376614 


•001788909 


560 


313600 


175616000 


23-6643191 


8-2425706 


•001785714 


561 


314721 


176558481 


23-6854386 


8-2474740 


•001782531 


562 


315844 


177504328 


23-7065392 


82523715 


•00,779359 


563 


316969 


178453547 


23-7276210 


8-2572633 


•001776199 


564 


318096 


179406144 


23-7486842 


8*2621492 


•001773050 


565 


319225 


180362125 


23-7697286 


8-2670294 


•00 1769912 


566 


320356 


181321496 


23-7907545 


8-2719039 


•001766784 


567 


321489 


182284263 


23-8117618 


8*2767726 


•001763668 


568 


322624 


183250432 


23-8327506 


8-2816355 


•001760563 


569 


323761 


184220009 


238537209 


8-2864928 


•001757469 


570 


324900 


185193000 


23-8746728 


8-2913444 


•001754386 


571 


326041 


186169411 


23-8956063 


8-2961903 


•001751313 


572 


327184 


187149248 


23-9165215 


8*3010304 


•001748252 


573 


328329 


188132517 


23-9374184 


8*3058651 


001745201 


574 


329476 


189119224 


23-9582971 


8-3106941 


•001742160 


575 


330625 


190109375 


23-9791576 


8-3155175 


•001739130 


576 


331776 


191102976 


24-0000000 


8*3203353 


•001736111 


577 


332929 


192100033 


24-0208243 


8-3251475 


•001733102 


578 


334084 


193100552 


24-0416306 


8-3299542 


•001730104 


579 


335241 


194104539 


24-0624188 


8-3347553 


•001727116 


580 


336400 


195112000 


240831891 


8-3395509 


•001724138 


581 


337561 


196122941 


24-1039416 


8-3443410 


•001721170 


582 


338724 


197137368 


24-1246762 


8-3491256 


•001718213 


583 


339889 


198155287 


241453929 


8-3539047 


•001715266 


584 


341056 


199176704 


24-1660919 


8-3586784 


•001712329 


585 


342225 


200201625 


24-1867732 


8-3634466 


•00i709402 


586 


343396 


201230056 


24-2074369 


8-3682095 


•001706485 


587 


344569 


202262003 


24-2280829 


8-3729668 


•001703578 


588 


345744 


203297472 


24-2487113 


8-3777188 


•001700680 


589 


346921 


204336469 


24-2693222 


8-3824653 


•001697793 


590 


348100 


205379000 


24-2899156 


8-3872065 


•001694915 


591 


349281 


206425071 


24-3104996 


8-3919423 


•001692047 


592 


350464 


207474688 


24-3310501 


8-3966729 


•001689189 


593 


351649 


208527857 


24-3515913 


8-4013981 


•001686341 


594 


352836 


209584584 


24-3721152 


8-4061180 


•001683502 


595 


354025 


210644875 


24-3926218 


8-4108326 


•0016806 T2 


596 


355216 


211708736 


24-4131112 


8-4155419 


•001677852 


597 


356409 


212776173 


24-4335834 


8-4202460 


•001675042 


598 


357604 


213847192 


24-4540385 


8-4249448 


•001672241 


599 


358801 


214921799 


24-4744765 


8-4296383 


•001669449 


600 


360000 


216000000 


24-4948974 


8-4343267 


•001666667 


601 


361201 


217081801 


24-5153013 


8-4390098 


•001663894 


602 


362404 


218167208 


245356883 


8-4436S77 


•001661130 


603 


363609 


219256227 


24-5560583 


8-4483605 


•001658375 


604 


364816 


220348864 


24-5764115 


8-4530281 


•001655629 


605 


366025 


221445125 


24-5967478 


8-4576906 


•001652893 


606 


367236 


222545016 


24-6170673 


8-4623479 


•001650165 


607 


368449 


223648543 


24-6373700 


8-4670001 


•001647446 


608 


369664 


224755712 


24-6576560 


8-4716471 


•001644737 



172 



SQUARES, CUBES, ETC., OF NUMBERS. 



No. , 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocal*. 


609 


370881 


225866529 


24-6779254 


8-4762892 


•001642036 


610 


372100 


226981000 


24-6981781 


8-4809261 


•001639344 


611 


373321 


228099131 


24-7184142 


8-4855579 


•001636661 


612 


374544 


229220928 


24-7386338 


8-4901848 


•001633987 


613 


375769 


230346397 


24-7588368 


8-4948065 


•001631321 


614 


376996 


231475544 


24-7790234 


8-4994233 


•001628664 


615 


378225 


232608375 


24-7991935 


8-5040350 


•001626016 


616 


379456 


233744896 


24-8193473 


8-5086417 


•001623377 


617 


380689 


234885113 


24-8394847 


8-5132435 


•001620746 


618 


381924 


236029032 


24-8596058 


8-5178403 


•001618123 


619 


383161 


237176659 


24-8797106 


8-5224331 


•001615509 


620 


384400 


238328000 


24-8997992 


8-5270189 


•001612903 


621 


385641 


239483061 


24-9198716 


8-5310009 


•001610306 


622 


38(3884 


240641848 


24-9399278 


8 5361780 


•001607717 


623 


388129 


241804367 


24-9599679 


8-5407501 


•001605136 


624 


389376 


242970624 


24-9799920 


8-5453173 


•001602564 


625 


. 390625 


244140625 


25-0000000 


8-5498797 


•001600000 


626 


391876 


245314376 


25-0199920 


8-5544372 


•001597444 


627 


393129 


246491883 


25-0399681 


8-5589899 


•001594896 


628 


394384 


247673152 


25-0599282 


8-5635377 


•001592357 


629 


395641 


248858189 


25-0798724 


8-5680807 


•001589825 


630 


396900 


250047000 


25-0998008 


8-5726189 


•001587302 


631 


398161 


251239591 


251197134 


8-5771523 


•001584786 


632 


399424 


252435968 


25-1396102 


8-5816809 


•001582278 


633 


400689 


253636137 


251594913 


8-5862047 


•001579779 


634 


401956 


254840104 


25 1793566 


8-5907238 


•001577287 


635 


403225 


256047875 


25 1992063 


8-5952380 


•001574803 


636 


404496 


257259456 


25-2190404 


8-5997476 


•001572327 


637 


405769 


258474853 


25-2388589 


8-6042525 


•001569859 


638 


407044 


259694072 


25 2586619 


8-6087526 


•001567398 


639 


408321 


260917119 


25-2784493 


8-6132480 


•001564945 


640 


409600 


262144000 


25-2982213 


8-6177388 


•001562500 


641 


410881 


263374721 


25-3179778 


8-6222248 


•001560062 


642 


412164 


264609288 


25-3377189 


8-6267063 


•001557632 


643 


413449 


265847707 


25-3574447 


8-6311830 


•001555210 


644 


414736 


267089984 


25-3771551 


8-6356551 


•001552795 


645 


416125 


268336125 


25-3968502 


8-6401226 


•001550388 


646 


417316 


26958^136 


25-4165302 


8-6445855 


•001547988 


647 


418609 


270840023 


25-4361947 


8-6490437 


•001545595 


648 


419904 


272097792 


25-4558441 


8-0534974 


•001543210 


649 


421201 


273359449 


254754784 


8-6579465 


•001540832 


650 


422500 


274625000 


25-4950976 


8-6623911 


•001538462 


651 


423801 


275894451 


25-5147016 


8-6668310 


•00153(5098 


652 


425104 


277167808 


25-5342901 


8-0712665 


•001533742 


653 


426409 


278445077 


25-5538647 


8-6756974 


•001531394 


654 


427716 


279726264 


25-5734237 


8-6801237 


•001529052 


655 


420025 


281011375 


25-5929678 


8-6845456 


•001526718 


656 


430336 


282300416 


25-6124969 


8-6889(530 


•001524390 


057 


431639 


283593393 


25-6320112 


8-6933759 


•001522070 


658 


432964 


284890312 


25-6515107 


8-6977843 


•001519757 


659 


434281 


286191179 


256709953 


8-7021882 


•00151"/451 


660 


435600 


287496000 


256904652 


8-7065877 


•001515152 


661 


436921 


288804781 


25-7099203 


8-7109827 


•001512859 


662 


438244 


290117528 


257293(507 


8-7153734 


•001510574 


663 


439569 


291434247 


25-74878(54 


8-7197596 


•001508296 


664 


440896 


292754944 


257681975 


8-7241414 


•00150(5024 


665 


442225 


294079(525 


257875939 


8-7285187 


001503759 


666 


443556 


29540829(5 


2580(59758 


87328918 


001501502 


667 


444889 


29(57409(53 


258263431 


8-7372004 


•001499250 


668 


446224 


298077(532 


25-845(59(50 


8-741(5246 


•001497006 


669 


447561 


299418309 


258650343 


87459846 


.001494768 


670 


448900 


300763000 


258843582 


8-7503401 


001492537 



SQUARES, CUBES, ETC., OF NUMBERS. 



173 



No. 


1 Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


671 


450241 


302111711 


25-9036677 


8-7546913 


•001490313 


672 


451584 


303464448 


259229628 


8-7590383 


•001488095 


673 


452929 


304821217 


259422435 


8-7633809 


•001485884 


674 


454276 


306182024 


25-9615100 


8-7677192 


•001483680 


675 


455625 


307546875 


25-9807621 


8-7720532 


•001481481 


676 


456976 


308915776 


26-0000000 


8-7763830 


•001479290 


677 


458329 


310288733 


280192237 


8-7807084 


•001477105 


678 


459684 


311665752 


26-6384331 


8-7850296 


•001474926 


679 


461041 


313046839 


26-0576284 


8-7893468 


•001472754 


680 


462400 


314432000 


26-0708096 


8-7936593 


•001470588 


681 


463761 


315821241 


260959767 


8-7979679 


•001468429 


682 


465124 


317214568 


261151297 


8-8022721 


•001466276 


683 


466489 


318611987 


26-1342687 


8-8065722 


•001464129 


684 


467856 


320013504 


26-1533937 


8-8108681 


•001461988 


685 


469225 


321419125 


26-1725047 


8-8151598 


•001459854 


686 


470596 


322828856 


26-1916017 


8-8194474 


•001457726 


68? 


471969 


324242703 


26-2106848 


8-8237307 


•001455604 


688 


473344 


325660672 


26-2297541 


8-8280099 


•001453488 


689 


474721 


327082709 


26-2488095 


8-8322850 


•001451379 


690 


476100 


328509000 


26-2678511 


8-8365559 


•001449275 


691 


477481 


329939371 


26-2868789 


8-8408227 


•001447178 


692 


478864 


331373888 


20-3058929 


8-8450854 


•001445087 


693 


480249 


332812557 


26-3248932 


8-8493440 


•001443001 


694 


481636 


334255384 


26-3438797 


8-8535985 


•001440922 


695 


483025 


335702375 


26-3628527 


8-8578489 


•001438849 


698 


484416 


337153536 


26-3818119 


8-8020952 


•001436782 


697 


485809 


338608873 


26-4007576 


8-8683375 


•001434720 


698 


487204 


3400G8392 


26-4196896 


8-8705757 


•001432665 


699 


488601 


341532099 


26-4386081 


8-8748099 


•001430615 


700 


490000 


343000000 


26-4575131 


8-8790400 


•001428571 


701 


491401 


344472101 


26-4764046 


8-8832061 


•001426534 


702 


492804 


345948408 


26-4952826 


8-8874882 


•001424501 


703 


494209 


347423927 


26-5141472 


8-8917063 


•001422475 


704 


495016 


348913664 


26-5329983 


8-8959204 


•001420455 


705 


497025 


350402625 


26-5518361 


8-9001304 


•001418440 


706 


498436 


351895816 


26-5706605 


8-9043366 


•001416431 


707 


499849 


353393243 


26-5894716 


8-9085387 


•001414427 


708 


501264 


354894912 


26-6082694 


8-9127369 


•001412429 


709 


502681 


356400829 


26-6270539 


8-9169311 


•001410437 


710 


504100 


357911000 


26-6458252 


8-9211214 


•001408451 


711 


505521 


359425431 


26-6645833 


8-9253078 


•001406470 


712 


506944 


360944128 


26-6833281 


89294902 


•001404494 


713 


508369 


362467097 


26-7020598 


8-9336687 


•001402525 


714 


509796 


363994344 ' 


26-7207784 


8-9378433 


•001400560 


715 


511225 


365525875 


26-7394839 


8-9420140 


•001398601 


716 


512656 


367061696 


26-7581763 


8-9461809 


001396648 


717 


514089 


368601813 


26-7768557 


8-9503438 


001394700 


718 


515524 


370146232 


26-7955220 


8-9545029 


•001392758 


719 


516961 


371694959 


26-8141754 


8-9586581 


•001390821 


720 


518400 


373248000 


26-8328157 


8-9628095 


•001388889 


721 


519841 


374805361 


26-8514432 


8-9669570 


•001386963 


722 


521284 


376367048 


26-8700577 


8-9711007 


001385042 


723 


522729 


377933067 


26-8886593 


8-9752406 


001383126 


724 


524176 


379503424 


26-9072481 


8-9793766 


•001381215 


725 


525625 


381078125 


26-9258240 


8-9835089 


001379310 


726 


527U76 


382657176 


26-9443872 


8-9876373 


•001377410 


727 


528529 


, 384240583 


26-9629375 


89917620 


001375516 


728 • 


529984 


385828352 


26-9814751 


8-9958899 


001373626 


729 


531441 


387420489 


27-0000000 


90000000 


•001371742 


730 


532900 


389017000 


270185122 


90041134 


•001369863 


731 


- 534361 


390617891 


270370117 


9-0082229 


•001367989 


732 


535821 


392223168 


270554985 


90123288 


001366120 



174 



SQUARES, CUBES, ETC., OF NUMBERS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


733 


537289 


393832837 


270739727 


9 0164309 


001364256 


734 


538756 


395446904 


27 0924344 


9 0205293 


•001362398 


735 


540225 


397065375 


271108834 


90246239 


•001360544 


736 


541696 


398688256 


271293199 


90287149 


•001358696 


737 


543169 


400315553 


271477439 


90328021 


•CJ1356852 


738 


544644 


401947272 


271661554 


90368857 


001355014 


739 


546121 


403583419 


271845544 


9 0409655 


001353180 


740 


547600 


405224000 


27-2029410 


9 0450419 


001351351 


741 


549081 


406869021 


27 2213152 


9 0491142 


001349528 


742 


550564 


408518488 


27-2396769 


90531831 


001347709 


743 


552049 


410172407 


27-2580263 


9 0572482 


001345895 


744 


553536 


411830784 


27-2763634 


90613098 


•001344086 


745 


555025 


413493625 


27-2946881 


9 0653677 


001342282 


746 


556516 


415160936 


27-3130006 


90694220 


001340483 


747 


558009 


416832723 


27-3313007 


9 0734726 


•001338688 


748 


559504 


418508992 


27-3495887 


9 0775197 


001336898 


749 


561001 


420189749 


273678644 


90815631 


001335113 


750 


562500 


421875000 


27-3861279 


9 0856030 


001333333 


751 


564001 


423564751 


27-4043792 


90890392 


001331558 


752 


565504 


425259008 


274226184 


9-0936719 


001329787 


753 


567009 


426957777 


27 4408455 


9 0977010 


•001328021 


754 


568516 


428661064 


27-4590604 


91017265 


•001326260 


755 


570025 


430368875 


274772633 


91057485 


001324503 


756 


571536 


432081216 


27-4954542 


91097669 


001322751 


757 


573049 


433798093 


275136330 


9-1137818 


•00132K304 


758 


574564 


435519512 


27-5317998 


9 1177931 


•001319261 


759 


576081 


437245479 


27 5499546 


9 1218010 


•001317523 


760 


577600 


438976000 


27-5680975 


9 1258053 


001315789 


761 


579121 


440711081 


27-5862284 


9 1298061 


001314060 


762 


580644 


442450728 


27 6043475 


91338034 


001312336 


763 


582169 


444194947 


27 6224546 


91377971 


•001310616 


764 


583696 


445943744 


27-6405499 


91417874 


•001308901 


765 


585225 


447697125 


27 6586334 


91457742 


001307190 


766 


586756 


449455096 


27-6767050 


91497576 


•001305483 


767 


588289 


451217663 


27-6947648 


91537375 


•001303781 


768 


589824 


452984832 


27-7128129 


91577139 


•001302083 


769 


591361 


454756609 


27 7308492 


9 1616869 


•001300390 


770 


592900 


456533000 


27-7488739 


91656565 


•001298701 


771 


594441 


458314011 


277668868 


91696225 


•001297017 


772 


595984 


460099648 


27-7848880 


9 1735852 


•001295337 


773 


597529 


461889917 


27 8028775 


91775445 


•001293061 


774 


599076 


463684824 


27-8208555 


9 1815003 


•001291990 


775 


600625 


465484375 


27 8388218 


91854527 


•001290323 


776 


602176 


467288576 


27 8567766 


91894018 


•001288660 


777 


603729 


469097433 


278747197 


9 1933474 


•001287001 


778 


605284 


470910952 


27-89265 J 4 


9 1972897 


•001285347 


779 


606841 


472729139 


27-9105715 


9 2012286 


•001283697 


780 


608400 


474552000 


27-9284801 


9 2051641 


•001282051 


781 


609961 


476379541 


27-9463772 


9 2090962 


•001280410 


782 


611524 


478211768 


27-9642629 


9 2130250 


•001278772 


783 


613089 


480048687 


27 9821372 


9 2169505 


•001277139 


784 


614656 


481890304 


280000000 


9 2208726 


001275510 


785 


616225 


483736625 


280178515 


9 2247914 


•001273885 


786 


617796 


485587656 


280356915 


9 2287068 


001272265 


787 


619369 


487443403 


280535203 


9 2326189 


•001270648 


788 


620944 


489303872 


28(1713377 


9 2365277 


001269036 


789 


622521 


491169069 


280891438 


9-2404333 


001267427 


790 


624100 . 


493039000 


281069386 


9 2443355 


001265823 


791 


625G81 


494913671 


281247222 


92482344 


001264223 


792 


627264 


496793088 


28 1424946 


9 2521300 


•001262626 


793 


628849 


498677257 


281602557 


92560224 


•001261034 


794 


1 630436 


1 500566184 


38 1780056 


92599114 


•001259446 



SQUARES, CUBES, ETC., OF NUMBERS. 



175 



* 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


795 


632025 


502459875 


281957444 


92637973 


001257862 


796 


633616 


504358336 


282134720 


9-2676798 


001256281 


797 


635209 


506261573 


282311884 


92715592 


001254705 


798 


636804 


508169592 


282488938 


92754352 


001253133 


799 


638401 


510082399 


28-2665881 


92793081 


001251364 


800 


640000 


512000000 


28 2842712 


92831777 


001250000 


SOI 


641601 


513922401 


28 3019434 


92870444 


001248439 


802 


643204 


515849608 


28-3196045 


92909072 


001246883 


803 


644809 


517781627 


283372546 


9 2947671 


001245330 


804 


646416 


519718464 


28-3548938 


92986239 


001243781 


805 


648025 


521660125 


283725219 


93024775 


001242236 


806 


649636 


523606616 


283901391 


93063278 


001240695 


807 


651249 


525557943 


28 4077454 


93101750 


001239157 


808 


652864 


527514112 


284253408 


93140190 


001237624 


809 


654481 


529475129 


284429253 


93178599 


•001236094 


810 


656100 


531441000 


28-4604989 


93216975 


001234568 


811 


657721 


533411731 


28-4780617 


93255320 


001233046 


812 


659344 


535387328 


284956137 


93293634 


•001231527 


813 


660969 


537367797 


28-5131549 


93331916 


001230012 


814 


662596 


539353144 


28-5306852 


93370167 


•001228501 


815 


664225 


541343375 


28-5482048 


93408386 


001226994 


816 


665856 


543338496 


28-5657137 


93446575 


001225490 


817 


667489 


545338513 


28-5832119 


93484731 


001223990 


818 


669124 


547343432 


28 6006993 


93522857 


001222494 


819 


670761 


549353259 


28 6181760 


93560952 


•001221001 


820 


672400 


551368000 


28 6356421 


93599016 


001219512 


821 


674041 


553387661 


28-6530976 


93637049 


001218027 


82-^ 


675684 


555412248 


286705424 


93675051 


001216545 


823 


677329 


557441767 


28 6879766 


9 3713022 


•001215067 


824 


678976 


559476224 


28-7054002 


93750963 


001213592 


825 


680625 


561515625 


28-7228132 


93788873 


001212121 


826 


682276 


563559976 


287402157 


93826752 


001210654 


827 


683929 


565609283 


28-7576077 


9-3864600 


001209190 


82S 


685584 


567663552 


28-7749891 


93902419 


001207729 


829 


687241 


569722789 


28-7923601 


93940206 


001206273 


830 


688900 


571787000 


28-8097206 


93977964 


001204819 


831 


690561 


573856191 


28-8270706 


9-4015691 


001203369 


832 


692224 


575930368 


28-8444102 


94053387 


001201923 


833 


693889 


578009537 


28-8617394 


94091054 


001200480 


834 


695556 


580093704 


28-8790582 


9-4128690 


001199041 


835 


697225 


582182875 


28-8963666 


9-4166297 


001197605 


836 


698896 


584277056 


289136646 


9-4203873 


001196172 


837 


700569 


586376253 


289309523 


9-4241420 


001194743 


838 


702244 


588480472 


28-9482297 


9-4278936 


001193317 


839 


703921 


590589719 


28 9654967 


9-4316423 


•001191895 


840 


705600 


592704000 


28 9827535 


9-4353880 


•001190476 


841 


707281 


594823321 


290000000 


9-4391307 


001189061 


842 


708964 


596947688 


290172363 


9-4428704 


001187648 


843 


710649 


599077107 


290344623 


9-4466072 


001186240 


844 


712336 


601211584 


290516781 


9-4503410 


001184834 


845 


714025 


603351125 


290688837 


9-4540719 


001183432 


846 


715716 


605495736 


290860791 


94577999 


•0011S2033 


847 


717409 


607645423 


291032644 


9-4615249 


•001180638 


848 


719104 


609800192 


291204396 


94652470 


001179245 


849 


720801 


611960049 


291376046 


94689661 


001177856 


850 


722500 


614125000 


291547595 


94726824 


001176471 


851 


724201 


616295051 


291719043 


94763957 


•001175088 


852 


725904 


618470208 


29 1890390 


9 4801061 


•001173709 


853 


727609 


620650477 


29 2061637 


94838136 


001172333 


854 


729316 


622835864 


29-2232784 


94875182 


•001170960 


855 


731025 


625026375 


292403830 


94912200 


001169591 


856 


732736 


627222016 


29-2574777 


94949188 


001168224 



176 



SQUARES, CUBES, ETC., OF NUMBERS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


857 


734449 


629422793 


292745623 


9-4986147 


•001166861 


858 


736164 


631628712 


29-2916370 


9-5023078 


•001165501 


859 


737881 


633839779 


29 3087018 


9-5059980 


•001164144 


860 


739600 


636056000 


29-3257566 


9-5096854 


•001162791 


861 


741321 


638277381 


29-3428015 


9-5133699 


•00116i440 


862 


743044 


640503928 


29-3598365 


9-5170515 


•001160093 


863 


744769 


642735647 


29-3768616 


9-5207303 


•001158749 


864 


746496 


644972544 


29-3938769 


9-5244063 


•001 157407 


865 


748225 


647214625 


29-5108823 


9-5280794 


•001156069 


866 


749956 


649461896 


29-4278779 


9-5317497 


•001154734 


867 


751689 


651714363 


294448637 


9-5354172 


•001153403 


868 


753424 


653072032 


29-4618397 


9-5390818 


•001152074 


869 


755161 


656234909 


29-4788059 


95427437 


•001150748 


870 


756900 


658503000 


29-4957624 


9-5464027 


•001149425 


871 


758641 


660776311 


29-5127091 


9-5500589 


•001148106 


872 


760384 


663054848 


29-5296461 


9 5537123 


•001146789 


873 


762129 


665338617 


29-5465734 


9-5573630 


•001145475 


874 


763876 


667627624 


29-5634910 


9-5610108 


•001144165 


875 


765625 


669921875 


29-5803989 


9-5646559 


•001142857 


876 


767376 


672221376 


29-5972972 


9-5682782 


•001141553 


877 


769129 


674526133 


29-6141858 


95719377 


•001140251 


878 


770884 


676836152 


29-6310648 


9 5755745 


•001138952 


879 


772641 


679151439 


29-6479342 


9-5792085 


001137656 


880 


774400 


681472000 


29 6647939 


95828397 


001136364 


881 


776161 


683797841 


29 6816442 


9 5864682 


•001135074 


882 


777924 


686128968 


29-6984848 


9-5900939 


001133787 


883 


779689 


688465387 


29-7153159 


95937169 


001132503 


884 


781456 


690807104 


29-7321375 


9-5973373 


001131222 


885 


783225 


693154125 


29-7489496 


96009548 


•001129944 


886 


784996 


695506456 


29-7657521 


9-6045696 


■001128668 


887 


786769 


697864103 


29-7825452 


9-6081817 


001127396 


888 


788544 


700227072 


29-7993289 


9 «J 17911 


001126126 


889 


790321 


702595369 


29-8161030 


96153977 


001124859 


890 


792100 


704969000 


29-8328678 


96190017 


001123596 


891 


793881 


707347971 


29-8496231 


9-6226030 


001122334 


892 


795664 


709732288 


29-8663690 


9-6262016 


001121076 


893 


797449 


712121957 


29-8831056 


9 6297975 


•001119821 


894 


799236 


714516984 


29-8998328 


9 6333997 


•001118568 


895 


801025 


716917375 


29-9165506 


9 6369812 


•001117818 


896 


802816 


719323136 


29-9332591 


9-6405690 


•001116071 


897 


804609 


721734273 


29-9499583 


96441542 


•001114827 


898 


806404 


724150792 


29-9666481 


96477367 


•001113586 


899 


808201 


726572699 


299833287 


9-6513166 


•001112347 


900 


810000 


729000000 


30-0000000 


9 6548938 


•00111111 1 


901 


811801 


731432701 


300166621 


9-6584684 


•001 109878 - 


902 


813604 


733870808 


300333148 


9-6620403 


•001 108647 


903 


815409 


736314327 


300499584 


9-6650096 


•001107420 


904 


817216 


738763264 


30-0665928 


9-6691762 


•001106195 


905 


819025 


741217625 


300832179 


96727403 


•001104972 


906 


820836 


743677416 


30-0998339 


9-6763017 


•001103753 


907 


822649 


746142643 


30 1164407 


9-6798604 


•001 102536 


908 


824464 


748613312 


301330383 


96834166 


001101322 


909 


826281 


751089429 


30-1496269 


9-686<r701 


•001 1001 10 


910 


828100 


753571000 


301662063 


9-6905211 


•001 09890 i 


911 


829921 


756058031 


301827765 


9-6940694 


•001097695 


912 


831744 


758550528 


30 1993377 


96976151 


•001096491 


913 


833569 


761048497 


30-2158899 


9-7011583 


•00 1095290 


914 


835396 


763551944 


302324329 


9-7046989 


•001094092 


915 


837225 


766060875 


302489669 


9-7082369 


001092896 


916 


839056 


768575296 


30 2654919 


9-7117723 


•001091703 


917 


840889 


771095213 


30-2820079 


9-7153051 


001090513 


918 


842724 


773620632 


30-2985148 


97188354 


•001089325 



SQUARES, CUBES, ETC., OF NUMBERS. 



171 



No. 


Squares. 


Cubes. 


Square Roots. 


Oube Roots, 


Reciprocals. 


919 


844561 


776151559 


303150128 


9-7223631 


•001088139 


920 


846400 


778688000 


303315018 


9-7258883 


•001086957 


921 


848241 


781229961 


30-3479818 


9-7294109 


•001085776 


922 


850084 


783777448 


30-3644529 


9-7329309 


•001084599 


923 


851929 


786330467 


303809151 


9-7364484 


•001083423 


924 


853776 


788889024 


30-3973683 


9-7399634 


•001082251 


925 


855625 


791453125 


304138127 


9-7434758 


•001081081 


926 


857476 


794022776 


30-4302481 


9-7469857 


•001079914 


927 


859329 


796597983 


30-4466747 


9-7504930 


•001078749 


928 


861184 • 


799178752 


30-4630924 


9-7539979 


•001077586 


929 


863041 


801765089 


30-4795013 


9-7575002 


•001076426 


930 


864900 


804357000 


304959014 


97610001 


•001075269 


931 


866761 


806954491 


30-5122926 


9-7644974 


•001074114 


932 


868624 


809557568 


30-5286750 


9-7679922 


•001072961 


933 


870489 


812166237 


30*5450487 


97714845 


•001071811 


934 


872356 


814780504 


305614136 


97749743 


•001070664 


935 


874225 


817400375 


30-5777697 


9-7784616 


•001069519 


936 


876096 


820025856 


30-5941171 


9-7819466 


•001068376 


937 


877969 


822656953 


306104557 


9-7854288 


•001067236 


938 


879844 


825293672 


30-6267857 


9-7889087 


•001066098 


939 


881721 


827936019 


306431069 


9-7923861 


•001064963 


940 


883600 


830584000 


30-6594194 


9-7958611 


•001063830 


941 


885481 


833237621 


30-6757233 


9-7993336 


•001062699 


942 


887364 


835896888 


30-6920185 


9-8028036 


•001061571 


943 


889249 


838561807 


30-7083051 


9-8062711 


•001060445 


944 


891136 


841232384 


30-7245830 


9-8097362 


•001059322 


945 


893025 


843908625 


307408523 


9-8131989 


•001058201 


946 


894916 


846590536 


30-7571130 


9-8166591 


•001057082 


947 


896809 


849278123 


307733651 


9-8201169 


•001055966 


948 


898704 


851971392 


30-7896086 


9-8235723 


•001054852 


949 


900601 


854670349 


308058436 


9-8270252 


•001053741 


950 


902500 


857375000 


30-8220700 


9-8304757 


•001052632 


951 


904401 


860085351 


30-8382879 


9-8339238 


001051525 


952 


906304 


862801408 


30-8544972 


9-8373695 


•001050420 


953 


908209 


865523177 


30-8706981 


9-8408127 


•001049318 


954 


910116 


.868250664 


30-8868904 


9-8442536 


•001048218 


955 


912025 


870983875 


309030743 


9-8476920 


•001047120 


956 


913936 


873722816 


30-9192497 


9-8511280 


•001046025 


957 


915849 


876467493 


30-9354166 


9-8545617 


•001044932 


958 


917764 


879217912 


30-9515751 


9-8579929 


•001043841 


959 


919681 


881974079 


309677251 


9-8614218 


•001042753 


960 


921600 


884736000 


30-9838668 


9-8648483 


•001041667 


961 


923521 


887503681 


31-0000000 


9-8682724 


•001040583 


962 


925444 


890277128 


310161248 


9-8716941 


•001039501 


963 


927369 


893056347 


310322413 


98751135 


•001038422 


964 


929296 


895841344 


310483494 


98785305 


•001037344 


965 


931225 


898632125 


310644491 


9-8819451 


•001036269 


966 


933156 


901428696 


31-0805405 


98853574 


•001035197 


967 


935089 


904231063 


310966236 


9-8887673 


•00103-4126 


968 


937024 


907039232 


311126984 


9-8921749 


•001033058 


969 


938961 


909853209 


31-1287648 


9-8955801 


•001031992 


970 


940900 


912673000 


311448230 


9-8989830 


•001030928 


971 


942841 


915498611 


311608729 


9-9023835 


•001029866 


972 


944784 


918330048 


31 1769145 


9-9057817 


•001028807 


973 


946729 


921167317 


3] 1929479 


9-9091776 


•001027749 


974 


948676 


924010424 


31-2089731 


9-91257)2 


•001026694 


975 


950625 


926859375 


312249900 


9-9159624 


•001025641 


976 


952576 


929714176 


312409987 


9-9193513 


•001024590 


977 


954529 


932574833 


31-2569992 


9-9227379 


•001023541 


978 


956484 


935441352 


312729915 


9-9261222 


•001022495 


979 


958441 


938313739 


31-2889757 


9-9295042 


•001021450 


980 


960400 


941192000 


313049517 


9-9328839 


001020406 



178 



SQ (TARES, CUBES, ETC., OF JSTDMBMiS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


981 


962361 


944076141 


31-3209195 


9-9362613 


001019168 


982 


964324 


946966168 


31-3338792 


9-9396363 


•001018330 


983 


966289 


949862087 


31-3528308 


9-9430092 


001017294 


984 


968256 


952763904 


31-3687743 


9-9463797 


•001016260 


985 


970225 


955671625 


31-3847097 


9-9497479 


•001015228 


986 


972196 


958585256 


31.4006369 


9-9531138 


•001014199 


987 


974169 


961504803 


31-4165561 


9-9564775 


•001013171 


988 


976144 


964430272 


31-4324673 


9-9598389 


•001012146 


989 


978121 


967361669 


31-4483704 


9-9631981 
9-9665549 


•001011122 


990 


980100 


970299000 


31-4642654 


•001010101 


991 


982081 


973242271 


31-4801525 


9-9699095 


•001009082 


992 


984064 


976191488 


31-4960315 


99732619 


•001008065 


993 


986049 


979146657 


31-5119025 


9-9766120 


•001007049 


994 


988036 


982107784 


31-5277655 


9-9799599 


•001006036 


995 


990025 


985074875 


31-5436206 


9-9833055 


001005025 


996 


992016 


988047936 


31-5594677 


9-9866488 


•001004016 


997 


994009 


991026973 


31-5753068 


9-9899900 


•001003009 


998 


996004 


994011992 


31-5911380 


9-9933289 


•001002004 


999 


998001 


997002999 


31-6069613 


9-9966656 


•001001001 


1000 


1000000 


1000000000 


31-6227766 


10-0000000 


•001000000 


1001 


1000201 


1003003001 


31-6385840 


100033222 


•0009990010 


1002 


1004004 


1006012008 


31-6543836 


10-0066622 


•0009980040 


1003 


1006009 


1009027027 


31.6701752 


10-0099899 


•0009970090 


1004 


1008016 


1012048064 


31-6859590 


100133155 


•0009960159 


1005 


1010025 


1015075125 


31-7017349 


100166389 


•0009950249 


1006 


1012036 


1018108216 


31-7175030 


100199601 


•0009940358 


1007 


1014049 


1021147343 


31-7332633 


100232791 


•0009930487 


1008 


1-016064 


1024192512 


31-7490157 


100265958 


•0009920635 


1009 


1018081 


1027243729 


31-7647603 


10-0299104 


-0009910803 


1010 


1020100 


1030301000 


31-7804972 


100332228 


•0009900990 


1011 


1022121 


1033364331 


31-7962262 


100365330 


•0009891197 


1012 


1(024144 


1036433728 


31-8119474 


100398410 


•0009881423 


1013 


1026169 


1039509197 


31-8276609 


100431469 


•0009871668 


1014 


1028196 


1042590744 


31-8433686 1 


100464506 


•0009861933 


1015 


1030225 


1045678375 


31-8590646 


100497521 


•0009852217 


1016 


1032256 


1048772096 


31-8747549 


100530514 


•0009842520 


1017 


1034289 


1051871913 


31-8904374 


10-0563485 


0009832842 


1018 


1036324 


1054977832 


31-9061123 


100596435 


0009823183 


1019 


1038361 


1058089859 


31-9217794 


100629364 


0009813543 


1020 


1040400 


1061208000 


31-9374388 


10-0668271 


0009803922 


1021 


1042441 


1064332261 


31-9530906 


100695156 


0009794319 


1022 


1044484 


1067462648 


31-9687347 


10-0728020 


0009784736 


1023 


1046529 


1070599167 


31-9843712 


10-0760863 


0009775171 


4024 


1048576 


1073741824 


320000000 


10-0793684 


0009765625 


1025 


1050625 


1076890625 


320156212 


10-0826484 


0009756098 


1026 


1052676 


1080045576 


320312348 


10-0859262 


0009746589 


1027 


1054729 


1083206683 


32 0468407 


10-0892019 


0009737098 


1028 


1056784 


1086373952 


320624391 


100924755 


0009727626 


1029 


1058841 


1089547389 


320780298 


100957469 


0009718173 


1030 


1060900 


1092727000 


32 0936131 


100990163 


•0009708732 


1031 


1062961 


1095912791 


321091887 


10-1022835 


•0009699321 


1032 


1065024 


1099104768 


32 1247568 


101055487 


•000968992? 


1033 


1067089 


1102302937 


321403173 


10-1088117 . 


•0009680542 


1034 


1069156 


1105507304 


321558704 


101120726 


•0009671180 


1035 


1071225 


1108717875 


321714159 


1011.53314 


•0009661836 


1036 


1073296 


1111934656 


32- 1869539 


10-1185882 


0009652510 


1037 


1075369 


1115157653 


32-2024844 


101218428 


0009643202 


1038 


1077444 


1 1 18386872 


32-2180074 


101250953 


0009633911 


1039 


1079521 


1121622319 


32 2335229 


101283457 


•0009624 635 


1040 


1081600 


1124864000 


32-2490310 


101315941 


0009615385 


1041 


1083681 


1128111921 


32-2645316 


10-1348403 


•0009606148 


1042 


1085764 


1131366088 


32-2800248 


101380845 


•0009596929 



TABLE XII. 

LOGARITHMS OF NUMBER! 
FROM 1 TO 10000. 



A 

TABLE, 

CONTAINING 

THE LOGARITHMS OF NUMBERS 

FROM 1 TO 10,000. 



NUMBERS FROM 1 TO 100 AND THEIR LOGARITHMS, '] 



WITH THEIR INDICES. 



No. 


Logarithm. 


No. 


Logarithm. 


No. 


Logarithm. 


No. 


Logarithm. 


No. 


Logarithm. 


1 


o-oooooo 


21 


1-322219 


41 


1-612784 


61 


1-785330 


81 


1908485 


2 


0-301030 


22 


1-342423 


42 


1-623249 


62 


1-792392 


82 


1-913814 


3 


0-477121 


23 


1-361728 


43 


1-633468 


63 


1-799341 


83 


1-919078 


4 


0-602060 


24 


1-38021 1 


44 


1-643453 


64 


1-806180 


84 


1-924279 


5 


0-698970 


25 


1-397940 


45 


1-653213 


65 


1-812913 


85 


1-929419 


6 


0-778151 


26 


1-414973 


46 


1-662758 


66 


1819544 


86 


1-934498 


7 


0-845098 


27 


1-431364 


47 


1-672098 


67 


1-826075 


87 


1-939519 


8 


0-903090 


28 


1-447158 


48 


1-681241 


68 


1-832509 


88 


1-944483 


9 


0-954243 


29 


1-462398 


49 


1-690196 


69 


1-838849 


89 


1-949390 


10 


1-000000 


30 


1-477121 


50 


1-698970 


70 


1-845098 


90 


1-954243 


11 


1041393 


31 


1-491362 


51 


1-707570 


71 


1-851258 


91 


1-959041 


12 


1079181 


32 


1-505150 


52 


1-716003 


72 


1-857332 


92 


1-963788 


13 


1113943 


33 


1-518514 


53 


1-724276 


73 


1-863323 


93 


1-968483 


14 


1146128 


34 


1-531479 


54 


1-732394 


74 


1 £69232 


94 


1-973128 


15 


1176091 


35 


1-544068 


55 


1-740363 


75 


1.875061 


95 


1-977724 


16 


1-204120 


36 


1-556303 


56 


1-748188 


76 


1-880814 


96 


1-982271 


17 


1-230449 


37 


1-568202 


57 


1-7.55875 


77 


1-886491 


97 


1-986772 


18 


1-255273 


38 


1-579784 


58 


1-763428 


78 


1-892095 


98 


1-991226 


19 


1-278754 


39 


1-591065 


59 


1-770852 


79 


1897627 


99 


1-995635 


20 


1-301030 


40 


1-602060 


60 


1-778151 80 


1-903090 


100 


2-000000 



Note. — In the following part of the Table, the characteristics are 
omitted, as they can be very easily supplied. Thus, the characteristic 
of the logarithm of every integer number, consisting only of one number, 
isO; of two figures, 1 ; of three figures, 2; of four figures, 3; being always 
a unit less than the number of figures contained in the integer number. 



180 



LOGARITHMS OF NUMBERS. 



181 



No. 


o I 


1 


» 


3 I 


4 I 


5 | 


6 I 


T 1 


8 


9 (Diff. 


100 


000000 


000434 


000868 


00 1301' 00 1734 002166 002598 


003029 


003461 


003891 432 


1 


4321 


4751 


5181 


5609 0038 0466 6894 


7321 


7748 


8174 428 


2 


8G00 


9026 


9451 


9876 01030O 010724 011147 


011570 


011993 


012415 424 


3 


012837 


013259 


013080 


014100; 4521 4940 


5360 


5779 


0197 


0616 420 


4 


7033 


7451 


7808 


8284 8700 9116 


9532 


9947 


020361 


020775 '416 


5 


021189 


021603 


022016 


022428 02284 1 023252 023664 


024075 


4486 


4890 412 


6 


5306 


5715 


6125 


0533 0942 7350 7757 


81G4 


8571 


8978 408 


7 


9384 


9789 


030195 


030000 031004 031408 031812 


032210 


032619 


033021 404 


8 


033424 


033826 


4227 


4(528 5029 


5430 


5830 


6230 


6629 


7028 400 


9 


7420 


7825 


8223 


8620 1 9017 


9414 


9811 


040207 


040602 


0409981397 


110 


041393 


041787 


042182 


042570 042909 043302 043755 


044148 


044540 


044932 393 


1 


5323 


5714 


6105 


6495; 6885 7275 7664 


8053 


8442 


8830 390 


2 


9218 


9606 


9993 


050380 050766 051153 051538 


051924 


052309 


052094 386 


3 


053078 


053163 


053846 


4230! 4613 


4996 


5378 


5760 


6142 


6524 383 


4 


6905 


7286 


7666 


8040 8426 


8805 


9185 


9563 


9942 


000320 379 


5 


060698 


061075 


061452 


061829 062206 062582 062958 


063333 


063709 


4083 376 


6 


4458 


4832 


5206 


5580 5953 6326 


6699 


7071 


7443 


7815 373 


7 


8186 


8557 


8928 


9298 9668 070038 


070407 


070776 


071145 


071514 370 


8 


071882 


072250 


072617 


072985 073352 


3718 


4085 


'1451 


4816 


5182 306 


9 


5547 


5912 


6276 


66401 7004 


7368 


7731 


8094 


8457 


8819 363 


120 


079181 


079543 


079904 


080266 080626 080987 


081347 


081707 


082067 


082426 360 


1 


082785 


083144 


083503 


3861 | 4219 


4576 


4934 


5291 


5647 


6004 357 


2 


6360 


6716 


7071 


7426; 7781 


8136 


8490 


8845 


9198 


9552 355 


3 


9905 


090258 


090011 


090963 091315 


091667 


092018 


092370 


092721 


093071 352 


4 


093422 


3772 


4122 


4471 


4820 


5169 


5518 


5866 


6215 


6562 349 


5 


6910 


7257 


7004 


7951 


8298 


8644 


8990 


9335 


9681 


100026 346 


6 


100371 


100715 


101059 


101403 


101747 


102091 


102434 


102777 


103119 


3462 343 


7 


3804 


4146 


4487 


4828 


5169 


5510 


5851 


6191 


6531 


6871 341 


8 


7210 


7549 


7888 


8227 


8565 


8903 


9241 


9579 


9916 


110253 338 


9 


11059U 


110026 


111203 


111599 


111934 


112270 


112505 


112940 


113275 


3609 335 


130 


1 13943 


114277 


114011 


114944 


115278 


115611 


115943 


116276 


116608 


116940 333 


1 


7271 


7603 


7934 


8265 


8595 


8926 


9256 


9586 


9915 


120245 330 


2 


120574 


120903 


121231 


121560 121888 


122216 


122544 


122871 


123198 


3525 328 


3 


3852 


4178 


4504 


4830 5156 


5481 


5806 


6131 


6456 


6781 325 


4 


7105 


7429 


7753 


8076] 8399 


-S722 


9045 


9368 


9690 


130012 323 


5 


130334 


130655 


130977 


131298 131619 


131939 


132260 


132580 


132900 


3219 321 


6 


3539 


3858 


4177 


4496 4814 


5133 


5451 


5769 


6086 


6403 318 


7 


6721 


7037 


7354 


7671 7987 


8303 


8618 


8934 


9249 


9564 316 


8 


9879 


140194 


140508 


140822 141136 


141450 


141763 


142076 


142389 


142702 314 


9 


143015 


3327 


3639 


3951 4263 


4574 


4885 


5196 


5507 


5818 1311 


140 


146128 


146438 


146748 


147058 147367 


147676 


147985 


148294 


148603 


148911 309 


1 


9219 


9527 


9835 


150142 150449 


150756 


151063 


151370 


151676 


151982 307 


2 


152288 


152594 


152900 


32051 3510 


3815 


4120 


4424 


4728 


5032 305 


3 


5336 


5640 


5943 


6246 6549 


6852 


7154 


7457 


7759 


8061 303 


4 


8362 


8604 


8965 


9266| 9567 


9868 


160168 


160469 


160709 


161068 301 


5 


161368 


161667 


161987 


162266 162564 


162863 


3161 


3460 


3758 


4055 299 


6 


4353 


4650 


4947 


5244J 5541 


5838 


6134 


6430 


6726 


7022 297 


7 


7317 


7613 


7908 


8203 8497 


8792 


9086 


9380 


9674 


9968 295 


8 


170262 


170555 


170848 


17U41 I 171434 


171726 


172019 


17231 1 


172603 


172895 293 


9 


3186 


3478 


3769 


4060 1 4351 


4641 


4932 


5222 


5512 


5802 291 


150 


176091 


176381 


176670 


176959' 177248 


177536 


177825 


178113 


178401 


178689 289 


1 


8977 


9204 


9552 


9839 


180126 


180413 


180699 


180986 


181272 


181558 287 


2 


181844 


182129 


182415 


182700 


2985 


3270 


3555 


3839 


4123 


4407 285 


3 


4691 


4975 


5259 


5542 


5825 


6108 


6391 


6674 


6956 


7239 283 


•1 


7521 


7803 


8084 


8366 8647 


8928 


9209 


9490 


9771 


190051 281 


5 


190332 


190012 


190892 


191171! 191451 


191730 


192010 


192289 


192567 


2846 279 


6 


3125 


3403 


3681 


3959 


4237 


4514 


4792 


5069 


5346 


5623 278 


7 


5900 


6176 


6453 


6729 


7005 


7281 


7556 


7832 


8107 


8382 276 


8 805 7 


8932 


9206 


9481 


9755 


200029 


200303 


200577 


200850 


201124 274 


9 201397 201670 


201943 


202216,202488 


2761 


3033 


3305 


3577 


3848 272 



No.| O L 1 I « 



4 I 



6 J 



9 | Diff. 



182 



LOGARITHMS OF NUMBERS. 



No. \ 


| 


1 1 


2 | 


3 | 


4 1 


5 I 


6 1 


7 I 


B I 


9 1 


Uiff 


160 


204120 


204391 


204663 


204934 


205204 


205475 205746 


206016 


206286 


206556)271 


1 


6826 


7096 


7365 


7634 


7904 


8173 


8441 


8710 


8979 


9247 


269 


2 


9515 


9783 


210051 


210319 


210586 


210853 


211121 


211388 


211654 


211921 


267 


3 


212188 


212454 


2720 


2986 


3252 


3518 


3783 


4049 


4314 


4579 


266 


4 


4844 


5109 


5373 


5638 


5902 


6166 


6430 


6694 


6957 


7221 


264 


5 


7484 


7747 


8010 


8273 


8536 


8798 


9060 


9323 


9585 


9846 


262 


6 


220108 


220370 


220631 


220892 


221153 


221414 


221675 


221936 


222196 


222456 


261 


7 


271o 


2976 


3236 


3496 


3755 


4015 


4274 


4533 


4792 


5051 


259 


8 


5300 


5568 


5826 


6084 


6342 


6600 


6858 


7115 


7372 


7630 


258 


9 


7887 


8144 


8400 


8657 


8913 


9170 


9426 


9682 


9938 


230193 


256 


170 


230449 


230704 


230960 


231215 


231470 


231724 


231979 


232234 


232488 


232742 


255 


1 


2990 


3250 


3504 


3757 


4011 


4264 


4517 


4770 


5023 


5276 


253 


2 


5528 


5781 


6033 


6285 


6537 


6789 


7041 


7292 


7544 


7795 


252 


3 


8046 


8297 


8548 


8799 


9049 


9299 


9550 


9800 


240050 


240300 


250 


4 


240549 


240799 


241048 


241297 


241546 


241795 


242044 


242293 


2541 


2790 


249 


5 


3038 


3286 


3534 


3782 


4030 


4277 


4525 


4772 


5019 


5266 


248 


6 


5513 


5759 


6006 


6252 


6499 


6745 


6991 


7237 


7482 


7728 


246 


7 


7973 


8219 


8464 


8709 


8954 


9198 


9443 


9687 


9932 


250176 


245 


8 


250420 


250664 


250908 


251151 


251395 


251638 


251881 


252125 


252368 


2610 


243 


9 


2853 


3096 


3338 


3580 


3822 


4064 


4306 


4548 


4790 


5031 


242 


180 


255273 


255514 


255755 


255996 


256237 


256477 


256718 


256958 


257198 


257439 


241 


1 


7679 


7918 


8158 


8398 


8637 


8877 


9116 


9355 


9594 


9833 


239 


2 


260071 


260310 


260548 


260787 


261025 


261263 


261501 


261739 


261976 


262214 


238 


3 


2451 


2688 


2925 


3162 


3399 


3636 


3873 


4109 


4346 


4582 


237 


4 


4818 


5054 


5290 


5525 


5761 


5996 


6232 


6467 


6702 


6937 


235 


5 


7172 


7406 


7641 


7875 


8110 


8344 


8578 


8812 


9046 


9279 


234 


6 


9513 


9746 


9980 


270213 


270446 


270679 


270912 


271144 


271377 


271609 


233 


7 


271842 


272074 


272306 


2538 


2770 


3001 


3233 


3464 


3696 


3927 


232 


8 


4158 


4389 


4620 


4850 


5081 


5311 


5542 


5772 


6002 


6232 


230 


9 


6462 


6692 


6921 


7151 


7380 


7609 


7838 


8067 


8296 


8525 


229 


190 


278754 


278982 


279211 


279439 


279667 


279895 


280123 


280351 


280578 


280806 


228 


1 


281033 


281261 


281488 


281715 


281942 


282169 


2396 


2622 


2849 


3075 


227 


2 


3301 


3527 


3753 


3979 


4205 


4431 


4656 


4882 


5107 


5332 


226 


3 


5557 


5782 


6007 


6232 


6456 


6681 


6905 


7130 


7354 


7578 


225 


4 


7802 


8026 


8249 


8473 


8696 


8920 


9143 


9366 


9589 


9812 


223 


5 


290035 


290257 


290480 


290702 


290925 


291147 


291369 


291591 


291813 


292034 


222 


6 


2256 


2478 


2699 


2920 


3141 


3363 


3584 


3804 


4025 


4246 


221 


7 


4466 


4687 


4907 


5127 


5347 


5567 


5787 


6007 


6226 


6446 


220 


8 


6665 


6884 


7104 


7323 


7542 


7761 


7979 


8198 


8416 


8635 


219 


9 


8853 


9071 


9289 


9507 


9725 


9943 


300161 


300378 


300595 


300813 


218 


209 


301030 


301247 


301464 


301681 


301898 


302114 


302331 


302547 


302764 


302980 


217 


1 


3196 


3412 


3628 


3844 


4059 


4275 


4491 


4706 


4921 


5136 


216 


2 


5351 


5566 


5781 


5996 


6211 


6425 


6639 


6854 


7068 


7282 


215 


3 


7496 


7710 


7924 


8137 


8351 


8564 


8778 


8991 


9204 


9417 


213 


4 


9630 


9843 


310056 


310268 


310481 


310693 


310906 


311118 


311330 


311542 


212 


5 


311754 


311966 


2177 


2389 


2600 


2812 


3023 


3234 


3445 


3656 


211 


6 


3867 


4078 


4289 


4499 


4710 


4920 


5130 


5340 


5551 


5760 


210 


7 


5970 


6180 


6390 


6599 


6809 


7018 


7227 


7436 


7646 


7854 


209 


8 


8063 


8272 


8481 


8689 


8898 


9106 


9314 


9522 


9730 


9938 


208 


9 


320146 


320354 


320562 


320769 


320977 


321184 


321391 


321598 


321805 


322012 


207 


210 


322219 


322426 


322633 


322839 


323046 


323252 


323458 


323665 


323871 


324077 


206 


1 


4282 


4488 


4694 


4899 


5105 


5310 


5516 


5721 


5926 


6131 


205 


2 


6336 


6541 


6745 


6950 


7155 


7359 


7563 


7767 


7972 


8176 


204 


3 


8380 


8583 


8787 


8991 


9194 


9398 


9601 


9805 


330008 


330211 


203 


4 


330414 


330617 


330819 


331022 


331225 


331427 


331630 


331832 


2034 


2236 


202 


5 


2438 


2640 


2842 


3044 


3246 


3447 


3649 


3850 


4051 


4253 


202 


6 


4454 


4655 


4850 


5057 


5257 


5458 


5658 


5859 


6059 


6260 


201 


7 


6460 


6660 


6860 


7060 


7260 


7459 


7659 


7858 


8058 


8257 


200 


8 


8456 


8656 


8855 


9054 


9253 


9451 


9650 


9849 


340047 


340246 


199 


9 


340444 


340642 


340841 


341039 


341237 


341435 


341632 


341830 


2028 


2225 


198 



No. | 



119 13 



* 1 5 



9 Dfl 



LOGARITHMS OF NUMBERS. 



183 



No.) 


I 


1 1 


2 1 


3 | 


4 1 


5 ! 


6 [ 


7 I 


8 I 


9 1 


DiS 


220 


342423 342620 


342817 


343014 


343212 


343409 


343606 


343802 


343999 


344196 


197 


1 


4392 


4589 


4785 


4981 


5178 


5374 


5570 


5766 


5962 


6157 


196 


2 


6353 


6549 


6744 


6939 


7135 


7330 


7525 


7720 


7915 


8110 


195 


3 


8305 


8500 


8694 


8889 


9083 


9278 


9472 


9666 


9860 


350054 


194 


4 


350248 


350442 


350636 


350829 


351023 


351216 


351410 


351603 


351796 


1989 


193 


5 


2183 


2375 


2568 


2761 


2954 


3147 


3339 


3532 


3724 


3916 


193 


6 


4108 


4301 


4493 


4685 


4876 


5068 


5260 


5452 


5643 


5834 


192 


7 


6026 


6217 


6408 


6599 


6790 


6981 


7172 


7363 


7554 


7744 


191 


8 


7935 


8125 


8316 


8506 


8696 


8886 


9076 


9266 


9456 


9646 


190 


9 


9835 


360025 


360215 


360404 


360593 


360783 


360972 


361161 


361350 


361539 


189 


130 


361728 


361917 


362105 


362294 


362482 


362671 


362859 


363048 


363236 


363424 


188 


1 


3612 


3800 


3988 


4176 


4363 


4551 


4739 


4926 


5113 


5301 


188 


2 


5488 


5675 


5862 


6049 


6236 


6423 


6610 


6796 


6983 


7169 


187 


3 


7356 


7542 


7729 


7915 


8101 


8287 


8473 


8659 


8845 


9030 


186 


4 


9216 


9401 


9587 


9772 


9958 


370143 


370328 


370513 


370698 


370883 


185 


5 


371068 


371253 


371437 


371622 


371806 


1991 


2175 


2360 


2544 


2728 


184 


6 


2912 


3096 


3280 


3464 


3647 


3831 


4015 


4198 


4382 


4565 


184 


7 


4748 


4932 


5115 


5298 


5481 


5664 


5846 


6029 


6212 


6394 


183 


8 


6577 


6759 


6942 


7124 


7306 


7488 


7670 


7852 


8034 


8216 


182 


9 


8398 


8580 


8761 


8943 


9124 


9306 


9487 


9668 


9849 


380030 


181 


240 


380211 


380392 


380573 


380754 


380934 


381115 


381296 


381476 


381656 


381837 


18} 


1 


2017 


2197 


2377 


2557 


2737 


2917 


3097 


3277 


3456 


3636 


180 


2 


3815 


3995 


4174 


4353 


4533 


4712 


4891 


5070 


5249 


5428 


179 


3 


5606 


5785 


5964 


6142 


6321 


6499 


6677 


6856 


7034 


7212 


178 


4 


7390 


7568 


7746 


7923 


8101 


8279 


8456 


8634 


8811 


8989 


178 


5 


9166 


9343 


9520 


9698 


9875 


390051 


390228 


390405 


390582 


390759 


177 


6 


390935 


391112 


391288 


391464 


391641 


1817 


1993 


2169 


2345 


2521 


176 


7 


2697 


2873 


3048 


3224 


3400 


3575 


3751 


3926 


4101 


4277 


176 


8 


4452 


4627 


4802 


4977 


5152 


5320 


5501 


5676 


5850 


6025 


175 


9 


6199 


6374 


6548 


6722 


6896 


7071 


7245 


7419 


7592 


7766 


174 


250 


397940 


398114 


398287 


398461 


398634 


398808 


398981 


399154 


399328 


399501 


173 


1 


9674 


9847 


400020 


400192 


400365 


400538 


400711 


400883 


401056 


40122S 


173 


2 


401401 


401573 


1745 


1917 


2089 


2261 


2433 


2605 


2777 


2949 


172 


3 


3121 


3292 


3464 3635 


3807 


3978 


4149 


4320 


4492 


4663 


171 


4 


4834 


5005 


5176 


5346 


5517 


5688 


5858 


6029 


6199 


6370 


171 


5 


6540 


6710 


6881 


7051 


7221 


7391 


7561 


7731 


7901 


8070 


170 


6 


8240 


8410 


8579 


8749 


8918 


9087 


9257 


9426 


9595 


9764 


169 


7 


9933 


410102 


410271 


410440 


410609 


410777 


410946 


411114 


411283 


411451 


169 


8 


411620 


1788 


1956 


2124 


2293 


2461 


2629 


2796 


2964 


3132 


168 


9 


3300 


3467 


3635 


3803 


3970 


4137 


4305 


4472 


4639 


4806 


167 


260 


414973 


415140 


415307 


415474 


415641 


415808 


415974 


416141 


416308 


416474 


167 


1 


C641 


6807 


6973 


7139 


7306 


7472 


7638 


7804 


7970 


8135 


166 


2 


8301 


8467 


8633 


8798 


8964 


9129 


9295 


9460 


9625 


9791 


165 


3 


9956 


420121 


4202S6 


420451 


420616 


420781 


420945 


421110 


421275 


421439 


165 


4 


421604 


1768 


1933 


2097 


2261 


2426 


2590 


2754 


2918 


3082 


164 


5 


3246 


3410 


3574 


3737 


3901 


4065 


4228 


4392 


4555 


4718 


164 


6 


4882 


5045 


5208 


5371 


5534 


5697 


5860 


6023 


6186 


6349 


163 


• 7 


6511 


6674 


6836 


6999 


7161 


7324 


7486 


7648 


7811 


7973 


162 


8 


8135 


8297 


8459 


8621 


8783 


8944 


9106 


9268 


9429 


9591 


162 


9 


9752 


9914 


430075 


430236 


430398 


430559 


430720 


430881 


431042 


431203 


161 


270 


431364 


431525 


431685 


431846 


432007 


432167 


432328 


432488 


432649 


432809 


161 


1 


2969 


3130 


3290 


3450 


3610 


3770 


3930 


4090 


4249 


4409 


160 


2 


4569 


4729 


4888 


5048 


5207 


5367 


5526 


5685 


5844 


6004 


159 


3 


6163 


6322 


6481 


6640 


6799 


6957 


7116 


7275 


7433 


7592 159 


4 


7751 


7909 


8067 


8226 


8384 


8542 


8701 


8859 


9017 


9175 


158 


5 


9333 


9491 


9648 


9806 


9964 


440122 


440279 


440437 


440594 


440752 


158 


e 


440909 


441066 


441224 


441381 


441538 


1695 


1852 


2009 


2166 


2323 


157 


7 


248Q 


2637 


2793 


2950 


3106 


3263 


3419 


3576 


3732 


3889 


157 


e 


404S 


4201 


4357 


4513 


4669 


4825 


4981 


5137 


5293 


5449 


156 


£ 


5604 


5760 


5915 


6071 


6226 


6382 


6537 


6692 


6848 


7003 


155 



No.|0|l|2i3|4|5l6 



8 | 9 iDiffi 



184 



LOGARITHMS OF NUMBERS. 



No. 


o 


1 


a 


3 


4 


5 


6 


7 


8 


280 


447158 


447313 447468 


447623 


447778 


447933 


448088 


448242 


448397 


1 


870G 


88G1 


9015 


9170 


9324 


9478 


9633 


9787 


9[)41 


2 


450249 


450403 


450557 


453711 


4508G5 


451018 


451172 


451326 


451479 


3 


1780 


1940 


2093 


2247 


2400 


2553 


2706 


2859 


3012 


4 


3318 


3471 


3624 


3777 


3930 


4082 


4235 


4387 


4540 


5 


4845 


4997 


5150 


5302 


5454 


5606 


5753 


5910 


6062 


C 


63GG 


6518 


6670 


6821 


6973 


7125 


7276 


7428 


7579 


7 


7882 


8033 


8184 


8336 


8487 


8638 


8789 


8940 


9091 


8 


9392 


9543 


9694 


9845 


9995 


460146 


460296 


460447 


460597 


9 


460898 


461048 


461198 


461348 


461499 


1649 


1799 


1948 


2098 


290 


462398 


462548 


462697 


462847 


462997 


46314G 


463296 


463445 


463594 


1 


3893 


4042 


4191 


4340 


4490 


4639 


4788 


4936 


5085 


2 5383 


5532 


56»0 


5829 


5977 


6126 


6274 


6423 


6571 


3 


6868 


7016 


7164 


7312 


7460 


7608 


7756 


7904 


8052 


4 


8347 


8495 


8643 


8790 


8938 


9085 


9233 


9380 


9527 


5 


9822 


9969 


470116 


470263 


470410 


470557 


470704 


470851 


470998 


6 


471292 


471438 


1585 


1732 


1878 


2025 


2171 


2318 


2464 


7 


2756 


2903 


3049 


3195 


3341 


3487 


2633 


3779 


3925 


8 


4216 


4362 


4508 


4653 


4799 


4944 


5090 


5235 


5381 


9 


5671 


5816 


5962 


6107 


6252 


6397 


6542 


6687 


6832 


300 


477121 


477266 


477411 


477555 


477700 


477844 


477989 


478133 


478278 


1 


8566 


8711 


8855 


8999 


9143 


9287 


9431 


9575 


9719 


2 


480007 


480151 


480294 


480438 


480582 


480725 


480869 


481012 


481156 


3 


1443 


1586 


1729 


1872 


2016 


2159 


2302 


2445 


2588 


4 


2874 


3016 


3159 


3302 


3445 


3587 


3730 


3872 


4015 


5 


4300 


4442 


4585 


4727 


4869 


5011 


5153 


5295 


5437 


6 


5721 


5863 


6005 


6147 


6289 


6430 


6572 


6714 


6855 


7 


7138 


7280 


7421 


7563 


7704 


7845 


7986 


8127 


8269 


8 


8551 


8692 


8833 


8974 


9114 


9255 


9396 


9537 


9677 


9 


9958 


490099 


490239 


490380 


490520 


490661 


490801 


490941 


491081 


310 


491362 


491502 


491642 


491782 


491922 


492062 


492201 


492341 


492-181 


1 


2760 


2900 


3040 


3179 


3319 


3458 


3597 


3737 


3876 


2 


4155 


4294 


4433 


4572 


4711 


4850 


4989 


5128 


5267 


3 


5544 


5683 


5822 


5960 


6099 


6238 


6376 


6515 


6653 


4 


6930 


7068 


7206 


7344 


7483 


7621 


7759 


7897 


8035 


5 


8311 


8448 


8586 


8724 


8862 


8999 


9137 


9275 


9412 


6 


9687 


9824 


9962 


500099 


500236 


500374 


500511 


500648 


500785 


7 


501059 


501 196 


501333 


1470 


1607 


1744 


1880 


2017 


2154 


8 


2427 


2564 


2700 


2837 


2973 


3109 


3246 


3382 


3518 


9 


3791 


3927 


4063 


4199 


4335 


4471 


4607 


4743 


4878 


320 


505150 


505286 


505421 


505557 


505693 


505828 


505964 


506099 


506234 


1 


6505 


6640 


6776 


6911 


7046 


7181 


7316 


7451 


7586 


2 


7856 


7991 


8126 


8260 


8395 


8530 


8664 


8799 


8934 


3 


9203 


9337 


9471 


9606 


9740 


9874 


510009 


510143 


510277 


4 


510545 


510679 


510813 


510947 


511081 


511215 


1349 


1482 


1616 


5 


1883 


2017 


2151 


2284 


2418 


2551 


2684 


2818 


2951 


6 


3218 


3351 


3484 


3617 


3750 


3883 


4016 


4149 


4282 


7 


4548 


4681 


4813 


4946 


5079 


5211 


5344 


5476 


5609 


8 


5874 


6006 


6139 


6271 


6403 


6535 


6668 


6800 


6932 


9 


7196 


7328 


7460 


7592 


7724 


7855 


7987 


3119 


8251 


330 


518514 


513646 


518777 


518909 


519040 


519171 


519303 


519434 


519566 


1 


9828 


9959 


520090 


520221 


520353 


520484 


520615 


520745 


520876 


2 


521138 


521269 


1400 


1530 


1661 


1792 


1922 


2053 


2183 


3 


2444 


2575 


2705 


2835 


2966 


3096 


3226 


3356 


3486 


4 


3746 


3876 


4006 


4136 


4266 


4396 


4526 


4656 


4785 


5 


5045 


5174 


5304 


5434 


5563 


5693 


5822 


5951 


6081 


6 


6339 


6469 


6598 


6727 


6856 


6985 


7114 


7243 


7372 


7 


7630 


7759 


7888 


8016 


8145 


8274 


8402 


8531 


8660 


8 


8917 


9045 


9174 


9302 


9430 


9559 


9687 


9815 


9943 


9 


530200 


530320 


530456 


530584 


530712 


530840 


530968 


531096 


531223 



9 | Diff; 



No. | O 



1|3I3|±|5|6J7|8|9 



I Dia 



LOGARITHMS OF NUMBERS. 



185 



lfo.| O 



a I 3 | 4 



6 ( 7 I 8/9| 



Diffi 

128 
127 
J27 
126 
126 
126 
125 
125 
125 
124 

124 
124 
123 
123 
123 
1-2-2 
122 
121 
12 ^ 
121 

120 
120 
120 
119 
119 
119 
119 
118 
118 
118 

117 
117 
117 
116 
116 
116 
115 
115 
115 
114 

114 
114 
114 
113 
113 
113 
112 
112 
112 
112 

111 
111 
111 

110 
110 
110 
110 
109 
109 
109 

3[3l4:|5|6|?jSl9|Diff 



340 ; 
1 

o 

3 
4 
5 

6 

7 
8 
9 

350 
1 
2 
3 
4 
5 
6 
7 
8 
9 

360 
1 
2 
3 
4 
5 
6 
7 



no 

1 
2 
3 
4 
5 
6 
7 



1 
2 
3 
4 
5 
6 
7 
8 
9 

390 



531479 


531607 


531734 


531802 


531990 


532117 


532245 


532372 


532500 


532627 


2754 


2882 


3009 


3130 


3264 


3391 


3518 


3645 


3772 


3899 


4026 


4153 


4280 


4407 


4534 


4601 


4787 


4914 


5041 


5167 


5294 


5421 


5547 


5674 


5800 


5927 


6053 


6180 


6306 


6432 


6558 


6685 


6811 


6937 


7063 


7189 


7315 


441 


7567 


7693 


7819 


7945 


8071 


8197 


8322 


8448 


8574 


8699 


8825 


8951 


9076 


9202 


9327 


9452 


9578 


9703 


9829 


9954 


540079 


540204 


540329 


540455 


540580 


540705 


540830 


540955 


541080 


541205 


1330 


1454 


1579 


1704 


1829 


1953 


2078 


2203 


2327 


2452 


2576 


2701 


2825 


2950 


3074 


3199 


3323 


3447 


3571 


3696 


3820 


3944 


544068 


544192 


544316 


544140 


544564 


544688 


544812 


544936 


545060 


545183 


5307 


5431 


5555 


5G78 


5802 


5925 


6049 


6172 


6296 


6419 


6543 


6066 


6789 


6913 


7036 


7159 


7282 


7405 


7529 


7652 


7775 


7898 


8021 


8144 


8267 


8389 


8512 


8635 


8758 


8881 


9003 


9126 


9249 


9371 


9494 


9616 


9739 


9861 


9984 


550106 


550228 


550351 


550473 


550595 


550717 


550840 


550962 


551084 


551206 


1328 


1450 


1572 


1694 


1816 


1938 


2060 


2181 


2303 


2425 


2547 


2668 


2790 


2911 


3033 


3155 


3276 


3398 


3519 


3640 


3762 


3883 


4004 


4126 


4247 


4368 


4489 


4610 


4731 


4852 


4973 


5094 


5215 


5336 


5457 


5578 


5699 


5820 


5940 


6061 


6182 


556303 


556423 


556544 


556664 


556785 


556905 


557026 


557146 


557267 


557387 


7507 


7627 


7748 


7868 


7988 


8108 


8228 


8349 


8469 


8589 


8709 


8829 


8948 


9068 


9188 


9308 


9428 


9548 


9667 


9787 


9907 


560026 


560146 


560265 


560385 


560504 


560624 


560743 


560863 


560982 


561101 


1221 


1340 


1459 


1578 


1698 


1817 


1936 


2055 


2174 


2293 


2412 


2531 


2650 


2769 


2887 


3006 


3125 


3244 


3362 


3481 


3600 


3718 


3837 


3955 


4074 


4192 


4311 


4429 


4548 


4666 


4784 


4903 


5021 


5139 


5257 


5376 


5494 


5612 


5730 


5848 


5966 


6084 


6202 


6320 


6437 


6555 


6673 


6791 


6909 


7026 


7144 


7262 


7379 


7497 


7614 


7732 


7849 


7967 


8084 


568202 


568319 


568436 


568554 


568671 


568788 


568905 


569023 


569140 


569257 


9374 


9491 


9608 


9725 


9842 


9959 


570076 


570193 


570309 


570426 


570543 


570660 


570776 


570893 


571010 


571 126 


1243 


1359 


1476 


1592 


1709 


1825 


1942 


2058 


2174 


2291 


2407 


2523 


2639 


2755 


2872 


2988 


3104 


3220 


3336 


3452 


3568 


3684 


3800 


3915 


4031 


4147 


4263 


4379 


4494 


_ 4610 


4726 


4841 


4957 


5072 


51G8 


5303 


5419 


5534 


5650 


5765 


5880 


5996 


6111 


6226 


6341 


6457 


6572 


6687 


6802 


6917 


7032 


7147 


7262 


7377 


7492 


7607 


7722 


7836 


7951 


8066 


8181 


8295 


8410 


8525 


8639 


8754 


8868 


8983 


9097 


9212 


9326 


9441 


9555 


9669 


579784 


579898 


580012 


580126 


580241 


580355 


580469 


580583 


580697 


580811 


580925 


581039 


1153 


1267 


1381 


1495 


1608 


1722 


1836 


1950 


2063 


2177 


2291 


2404 


2518 


2631 


2745 


2858 


2972 


3085 


3199 


3312 


3426 


3539 


3652 


3765 


3879 


3992 


4105 


4218 


4331 


4444 


4557 


4670 


4783 


4896 


5009 


5122 


5235 


5348 


5461 


5574 


5686 


5799 


5912 


6024 


6137 


6250 


6362 


6475 


6587 


6700 


6812 


6925 


7037 


7149 


7262 


7374 


7486 


7599 


7711 


7823 


7935 


8047 


8160 


8272 


8384 


8496 


8608 


8720 


8832 


8944 


9056 


9167 


9279 


9391 


9503 


9615 


9726 


9838 


9950 


590061 


590173 


590284 


590396 


590507 


590619 


590730 


590842 


590953 


591065 


591176 


591287 


591399 


591510 


591621 


591732 


591843 


591955 


592066 


2177 


2288 


2399 


2510 


2621 


2732 


2843 


2954 


3064 


3175 


3286 


3397 


35U8 


3618 


3729 


3840 


3950 


4061 


4171 


4282 


4393 


4503 


4614 


4724 


4834 


4945 


5055 


5165 


5276 


5386 


5496 


5006 


5717 


5827 


5937 


6047 


6157 


6267 


6377 


6487 


6597 


6707 


6817 


6927 


7037 


7146 


7256 


7366 


7476 


7586 


7695 


7805 


7914 


8024 


8134 


8243 


8353 


8462 


8572 


8681 


8791 


89C0 


9009 


9119 


9228 


9337 


9446 


9556 


9665 


9774 


9883 


9992 


600101 


600210 


€00319 


600428 


600537 


600646 


600755 


600864 


600973 


601082 


1191 


1299 


1408 


1517 


1625 


1734 


1843 


1951 



186 



LOGARITHMS OF NUMBERS. 



No. | [ 1 1 



|3|4|5|617|8|9 f Di« 



400 


6Q2060 


302169 


502277 


602386 


602494 


602603 


602711 


602819 


602928 603036 


108 


1 


3144 


3253 


3361 


3469 


3577 


3686 


3794 


3902 


4010 


4118 


108 


2 


4226 


4334 


4442 


4550 


4658 


4766 


4874 


4982 


5089 


5197 


108 


3 


5305 


5413 


5521 


5628 


5736 


5844 


5951 


6059 


6166 


6274 


108 


4 


6381 


6489 


6596 


6704 


6811 


6919 


7026 


7133 


7241 


7348 


107 


5 


7455 


7562 


7669 


7777 


7884 


7991 


8098 


8205 


8312 


8419 


107 


6 


8526 


8633 


8740 


8847 


8954 


9061 


9167 


9274 


9381 


9488 


107 


7 


9594 


9701 


9808 


9914 


610021 


610128 


610234 


610341 


610447 


610554 


107 


8 


610660 


510767 


610873 


610979 


1086 


1192 


1298 


1405 


1511 


1617 


106 


9 


1723 


1829 


1936 


2042 


2148 


2254 


2360 


2406 


2572 


2678 


106 


110 


612784 


612890 


612996 


613102 


613207 


613313 


613419 


613525 


613630 


613736 


106 


1 


3842 


3947 


4053 


4159 


4264 


4370 


4475 


4581 


4686 


4792 


106 


2 


4897 


5003 


5108 


5213 


5319 


5424 


5529 


5634 


5740 


5845 


105 


3 


5950 


6055 


6160 


6265 


6370 


6476 


6581 


6686 


6790 


6895 


105 


4 


7000 


7105 


7210 


7315 


7420 


7525 


7629 


7734 


7839 


7943 


105 


5 


8048 


8153 


8257 


8362 


8466 


8571 


8676 


8780 


8884 


8989 


105 


6 


9093 


9198 


9302 


9406 


9511 


9615 


9719 


9824 


9928 


620032 


104 


7 


620136 


620240 


620314 


620448 


620552 


620656 


620760 


620864 


620968 


1072 


104 


8 


1176 


1280 


1384 


1488 


1592 


1695 


1739 


1903 


2007 


2110 


104 


9 


2214 


2318 


2421 


2525 


2628 


2732 


2835 


2939 


3042 


3146 


104 


120 


023249 


623353 


623456 


623559 


623663 


623766 


623869 


623973 


624076 


624179 


103 


1 


4282 


4385 


4488 


4591 


4695 


4798 


4901 


5004 


5107 


5210 


103 


2 


5312 


5415 


5518 


5621 


5724 


5827 


5929 


6032 


6135 


6238 


103 


3 


6340 


6443 


6546 


6648 


6751 


6853 


6956 


7058 


7161 


7263 


103 


4 


7366 


7468 


7571 


7673 


7775 


7878 


7980 


8082 


8185 


8287 


102 


5 


8389 


8491 


8593 


8695 


8797 


8900 


9002 


9104 


9206 


9308 


102 


6 


9410 


9512 


9613 


9715 


9817 


9919 


630021 


630123 


630224 


630326 


102 


7 


630428 


630530 


630631 


630733 


630835 


630936 


1038 


1139 


1241 


1342 


102 


8 


1444 


1545 


1647 


1748 


1849 


1951 


2052 


2153 


2255 


2356 


101 


9 


2457 


2559 


2660 


2761 


2862 


2963 


3064 


3165 


3266 


3367 


101 


430 


633468 


633569 


633670 


633771 


633872 


633973 


634074 


634175 


634276 


634376 


101 


1 


4477 


4578 


4679 


4779 


4880 


4981 


5081 


5182 


5283 


5383 


101 


2 


5484 


5584 


5685 


5785 


5886 


5986 


6087 


6187 


6287 


6388 


100 


3 


6488 


6588 


6688 


6789 


6889 


6989 


7089 


7189 


7290 


7390 


100 


4 


7490 


7590 


7690 


7790 


7890 


7990 


8090 


8190 


8290 


8389 


100 


5 


8489 


8589 


8689 


8789 


8888 


8988 


9088 


9188 


9287 


9387 


100 


6 


9486 


9586 


9686 


9785 


9885 


9984 


640084 


640183 


640283 


640382 


99 


7 


640481 


640581 


640680 


640779 


640879 


640978 


1077 


1177 


1276 


1375 


99 


8 


1474 


1573 


1672 


1771 


1871 


1970 


2069 


2168 


2267 


2366 


99 


9 


2465 


2563 


2662 


2761 


2860 


2959 


3058 


3156 


3255 


3354 


99 


440 


643453 


643551 


643650 


643749 


643847 


643946 


644044 


644143 


614242 


644340 


98 


1 


4439 


4537 


4636 


4734 


4832 


4931 


5029 


5127 


5226 


5324 


98 


2 


5422 


5521 


5619 


5717 


5815 


5913 


6011 


6110 


6208 


6306 


98 


3 


6404 


6502 


6600 


6698 


6796 


6894 


6992 


7089 


7187 


7285 


98 


4 


7383 


7481 


7579 


7676 


7774 


7872 


7989 


8067 


8165 


8262 


98 


5 


8360 


8458 


8555 


8653 


8750 


8848 


8945 


9043 


9140 


9237 


97 


6 


9335 


9432 


9530 


9627 


9724 


9821 


9919 


650016 


650113 


650210 


97 


7 


650308 


650405 


650502 


650599 


650696 


650793 


650890 


0987 


1084 


1181 


97 


e 


1278 


1375 


1472 


1569 


1666 


1762 


1859 


1956 


2053 


2150 


97 


s 


2246 


2343 


2440 


2536 


2633 


2730 


2826 


2923 


3019 


3116 


97 


450 


653213 


653309 


653405 


653502 


653598 


653695 


653791 


653888 


653984 


654080 


96 


] 


4177 


4273 


4369 


4465 


4562 


4658 


4754 


4850 


4946 


5042 


96 


2 


5138 


5235 


5331 


5427 


5523 


5619 


5715 


5810 


5906 


6002 


96 


2 


\ 6098 


<>194 


6290 


6386 


6482 


6577 


6673 


6760 


6864 


6960 


96 


< 


7056 


7152 


7247 


7343 


7438 


7534 


7629 


7725 


782.) 


7916 


96 




► 801) 


8107 


8202 


8298 


8393 


8488 


8584 


8679 


8774 


8870 


95 


e 


► 89C5 


9060 


9155 


9250 


934C 


9441 


953C 


9631 


9726 


9821 


95 


7l 9916 


660011 


660106 


660201 


66029C 


660391 


660486 


660581 


660676 


660771 


95 


81660865 


0960 


1055 


1150 


1245 


1339 


1434 


1529 


1623 


1718 


95 


i 


l| J813 


1907 


2002 


2096 


2191 


2286 


23£C 


2475 


2569 


2663 


95 



Na.IOIll3l3l4rl5l6l7l8l9lD.il 



LOGARITHMS OF NUMBERS. 



187 



No. | 



3 | S | 4 | 5 | 6 | 7 



9 |Diff 



460 662758 


66-2852,662947 


663041 663135 663230 663324 663418 


663512 663607 


94 


1 


3701 


3795 


3889 


3983 


40781 


4172 


4266 


4360 


4454; 


4548 


94 


2 


4642 


4736 


4830 


4924 


5018 


5112 


5206: 


5299 


5393 


5487 


94 


3 


5581 


5675 


5769 


5862 


5956 


6050 


6143 1 


6237 


6331 


6424 


94 


4 


6518 


6612 


6705 


6799 


6892 


6986 


7079: 


7173 


7266 


7360 


94 


5 


7453 


7546 


7640 


7733 


7826 


7920 


8013 i 


8106 


8199 


8293 


93 


G 


8386 


8479 


8572 


8665 


8759 


8852 


89451 


9038 


9131 1 


9224 


93 


7 


9317 


9410 


9503 


9596 


9689 


9782 


9875! 


9967 


670060 


670153 


93 


8 


670246 


670339 


67043] 


670524 


670617 


670710 


670802 


670895 


0988 i 


1080 


93 


9 


1173 


1265 


1358 


1451 


1543 


1636 


1728J 


1821 


1913 


2005 


93 


470 


672098 


672190 


672283 


672375 


672467 


672560 


572652 ' 


672744 


672836 


672929 


92 


1 


3021 


3J13 


3205 


3297 


3390 


3482 


3574 


3666 


3758 


3850 


92 


2 


3942 


4034 


4126 


4218 


4310 


4402 


4494 


4586 


4677 


4769 


92 


3 


4861 


4953 


5045 


5137 


5228 


5320 


5412 


5503 


5595 


5687 


92 


4 


5778 


5870 


5962 


6053 


6145 


6236 


6328 


6419 


6511 


6602 


92 


5 


6694 


6785 


6876 


6968 


7059 


7151 


7242 


7333 


7424 


7516 


91 


6 


7607 


7698 


7789 


7881 


7972 


8063 


8154 


8245 


8336 


8427 


91 


7 


a - >i8 


8609 


8700 


8791 


8882 


8973 


9064 


9155 


9246 


9337 


91 


8 


9428 


9519 


9610 


9700 


9791 


9882 


9973 


680063 


680154 


680245 


91 


9 


680336 


680426 


680517 


68C607 


680698 


680789 


680879 


0970 


1060 


1151 


91 


480 


681241 


681332 


681422 


681513 


681603 


681693 


681784 


681874 


681964 


682055 


90 


1 


2145 


2235 


2326 


2416 


2506 


2596 


2686 


2777 


2£67 


2957 


90 


2 


3047 


3137 


3227 


3317 


3407 


3497 


3587 


3677 


3767 


3857 


90 


3 


3947 


4037 


4127 


4217 


4307 


4396 


4486 


4576 


4666 


4756 


90 


4 


4845 


4935 


5025 


5114 


5204 


5294 


5383 


5473 


5563 


5652 


90 


5 


5742 


5831 


5921 


6010 


6100 


6189 


6279 


6368 


6458 


6547 


89 


6 


6636 


6726 


6815 


6904 


6994 


7083 


7172 


7261 


7351 


7440 


89 


7 


7529 


7618 


7707 


7796 


7886 


7975 


8064 


8153 


8242 


8331 


89 


8 


8420 


8509 


8598 


8687 


8776 


8865 


8953 


9042 


9131 


9220 


89 


9 


9309 


9398 


9486 


9575 


9864 


9753 


9841 


9930 


690019 


690107 


89 


490 


690196 


690285 


690373 


690462 


690550 


690639 


690728 


690816 


690905 


690993 


89 


1 


1081 


1170 


1258 


1347 


1435 


1524 


1612 


1700 


1789 


1877 


88 


2 


1965 


2053 


2142 


2230 


2318 


2406 


2494 


2583 


2671 


2759 


88 


3 


2847 


2935 


3023 


3111 


3199 


3287 


3375 


3463 


3551 


3639 


88 


4 


3727 


3815 


3903 


3991 


4078 


4166 


4254 


4342 


4430 


4517 


88 


5 


4605 


4693 


4781 


4868 


4956 


5044 


5131 


5219 


5307 


5394 


88 


6 


5482 


5569 


5657 


5744 


583i 


5919 


6007 


6094 


6182 


6269 


87 


7 


6356 


6444 


6531 


6618 


6706 


6793 


6880 


6968 


7055 


7142 


87 


8 


7229 


7317 


7404 


7491 


7578 


7665 


7752 


7839 


7926 


8014 


87 


9 


8101 


8188 


8275 


8362 


8449 


8535 


8622 


8709 


8796 


8883 


87 


500 


698970 


699057 


699144 


699231 


699317 


699404 


699491 


699578 


699664 


699751 


87 


1 


9838 


9924 


700011 


700098 


700184 


700271 


700358 


700444 


700531 


700617 


87 


2 


700704 


700790 


0877 


0963 


1050 


1136 


1222 


1309 


1395 


1482 


86 


3 


1568 


1654 


1741 


1827 


1913 


1999 


2086 


2172 


2258 


2344 


86 


4 


2431 


2517 


2603 


2689 


2775 


2861 


2947 


3033 


3119 


3205 


86 


5 


3291 


3377 


3463 


3549 


3635 


3721 


3807 


3893 


3979 


4065 


86 


6 


4151 


4236 


4322 


4408 


4494 


4579 


4665 


4751 


4837 


4922 


86 


7 


5008 


5094 


5179 


5265 


5350 


5436 


5522 


5607 


5693 


5778 


86 


8 


5864 


5949 


6035 


6120 


6206 


6291 


6376 


6462 


6547 


6632 


85 


9 


6718 


6803 


6888 


6974 


7059 


7144 


7229 


7315 


7400 


7485 


85 


510 


707570 


707655 


707740 


707826 


707911 


707996 


708081 


708166 


708251 


708336 


85 


1 


8421 


8506 


8591 


8676 


8761 


8846 


8931 


9015 


9100 


9185 


85 


2 


9270 


9355 


9440 


9524 


9609 


9694 


9779 


9863 


9948 


710033 


85 


3 


710117 


710202 


710287 


710371 


710456 


710540 


710025 


710710 


710794 


0879 


85 


4 


0963 


1048 


1132 


1217 


1301 


1385 


1470 


1554 


1639 


1723 


84 


5 


1807 


1892 


1970 


2060 


2144 


2229 


2313 


2397 


2481 


2566 


84 


6 


2650 


2734 


2818 


2902 


2986 


3070 


3154 


3238 


3323 


3407 


84 


7 


3491 


3575 


3659 


3742 


3826 


3910 


3994 


4078 


1 4162 


4246 


84 


8 


4330 


4414 


449"J 


4581 


4665 


4749 


4833 


4916 


1 5000 


5084 


84 


9 


5167 


5251 


5335 


5418 


5502 | 5586 


5669 


5753 


I 5836 


J 592C 


84 



No. 



112)3i41316|7|8)9|Diff 



188 



LOGARITHMS OF NUMBERS. 



No. 


o 


1 


3 


3 


* 


5 1 


6 


7 


8 


r 9 


Diffi 


520 


716003 


716087 


716170 716254 


716337 


716421 


716504 


716588 


716671 


716754 


83 


1 


6838 


6921 


7004 


7088 


7171 


7254 


7338 


7421 


7504 


7587 


83 


2 


7671 


7754 


7837 


7920 


8003 


8086 


8169 


8253 


8336 


8419 


83 


3 


8502 


8585 


8668 


8751 


8834 


8917 


9000 


9083 


9165 


9248 


83 


4 


9331 


9414 


9497 


9580 


9663 


9745 


9828 


9911 


9994 


720077 


83 


5 


720159 


720242 


720325 


720407 


720490 


720573 


720655 


720738 


720821 


0903 


83 


6 


0986 


1068 


1151 


1233 


13i6 


1398 


1481 


1563 


1646 


1728 


82 


7 


18U 


1893 


1975 


2058 


2140 


2222 


2305 


2387 


2469 


2552 


82 


8 


2G34 


2718 


2798 


2881 


2963 


3045 


3127 


3209 


3291 


3374 


82 


9 


3456 


3538 


3620 


3702 


3784 


3866 


3948 


4030 


4112 


4194 


82 


530 


724276 


724358 


724440 


724522 


724604 


724685 


724767 


724849 


724931 


725013 


82 


1 


5095 


5176 


5258 


5340 


5422 


5503 


5585 


5667 


5748 


5830 


82 


o 


5912 


5993 


6075 


6156 


6238 


6320 


6401 


6483 


6564 


6646 


82 


3 


6727 


6809 


6890 


6972 


7053 


7134 


7216 


7297 


7379 


7460 


81 


4 


7541 


7623 


7704 


7785 


7866 


7948 


8029 


8110 


8191 


8273 


81 


5 


8354 


8435 


8516 


8597 


8678 


8759 


8841 


8922 


9003 


9084 


81 


6 


9165 


9246 


9327 


9408 


9489 


9570 


9651 


9732 


9813 


9893 


81 


7 


9974 


730055 


730136 


730217 


730298 


730378 


730459 


730540 


730621 


730702 


81 


8 


730782 


0863 


0944 


1024 


1105 


1186 


1266 


1347 


1428 


1508 


81 


Q 


1589 


1669 


1750 


1830 


1911 


1991 


2072 


2152 


2233 


2313 


81 


540 


732394 


732474 


732555 


732635 


732715 


732798 


732876 


732956 


733037 


733117 


80 


1 


3197 


3273 


3358 


3438 


3518 


3598 


3679 


3759 


3839 


3919 


80 


2 


3999 


4079 


4160 


4240 


4320 


4400 


4480 


4560 


4610 


4720 


80 


3 


4800 


4880 


4960 


5040 


5120 


5200 


5279 


5359 


5439 


5519 


80 


4 


5599 


5679 


5759 


5838 


5918 


5998 


6078 


6157 


6237 


6317 


80 


5 


6397 


6476 


6556 


6635 


6715 


6795 


6874 


6954 


7034 


7113 


80 


6 


7193 


7272 


7352 


7431 


7511 


7590 


7670 


7749 


7829 


7908 


79 


7 


7987 


8067 


8146 


8225 


8305 


8384 


8463 


8543 


8622 


8701 


79 


8 


8781 


8860 


8939 


9018 


9097 


9177 


9256 


9335 


9414 


9493 


79 


9 


9572 


9651 


9731 


9810 


9889 


9968 


740047 


740126 


740205 


740284 


79 


550 


710363 


740442 


740521 


740600 


740678 


740757 


740836 


740915 


740994 


741073 


79 


1 


1152 


1230 


1309 


1388 


1467 


1546 


1624 


1703 


1782 


1860 


79 


2 


1939 


2018 


2096 


2175 


2254 


2332 


2411 


2489 


2568 


2647 


79 


3 


2725 


2804 


2882 


2981 


3039 


3118 


3196 


3275 


3353 


3431 


78 


4 


3510 


3588 


3667 


3745 


3823 


3902 


3980 


4058 


4136 


4215 


78 


5 


4293 


4371 


4449 


4528 


4606 


4684 


4762 


4840 


4919 


4997 


78 


6 


5075 


5153 


5231 


5309 


5387 


5465 


5543 


5621 


5699 


5777 


78 


7 


5855 


5933 


6011 


6089 


6167 


6245 


6323 


6401 


6479 


6556 


78 


8 


6634 


6712 


6790 


6868 


6945 


7023 


7101 


7179 


7256 


7334 


78 


9 


7412 


7489 


7567 


7645 


7722 


7800 


7878 


7955 


8033 


8110 


78 


560 


748188 


748266 


748343 


748421 


748498 


748576 


748653 


748731 


748808 


748885 


77 


1 


8963 


9040 


9118 


9195 


9272 


9350 


9427 


9504 


9582 


9659 


77 


2 


9736 


9814 


9891 


9968 


750045 


750123 


750200 


750277 


750354 


750431 


77 


3 


750508 


750586 


750663 


750740 


0817 


0894 


0971 


1048 


1125 


1202 


77 


4 


1279 


1356 


1433 


1510 


1587 


1664 


1741 


1818 


1895 


1972 


77 


5 


2048 


2125 


2202 


2279 


2356 


2433 


2509 


2586 


2663 


2740 


77 


6 


2816 


2893 


2970 


3047 


3123 


3200 


3277 


3353 


3430 


3506 


77 


7 


3583 


3660 


3736 


3813 


3889 


3966 


4042 


4119 


4195 


4272 


77 


8 


4348 


4425 


4501 


4578 


4654 


4730 


4807 


4833 


4960 


5036 


76 


9 


5112 


5189 


5265 


5341 


5417 


5494 


5570 


5646 


5722 


5799 


76 


570 


755875 


755951 


756027 


756103 


756180 


756256 


756332 


756408 


756484 


756560 


76 


I 


6636 


6712 


6788 


6864 


6940 


7016 


7092 


7168 


7244 


7320 


78 


2 


7396 


7472 


7548 


7624 


7700 


7775 


7851 


7927 


8003 


8079 


76 


3 


8155 


8230 


8306 


8382 


8458 


8533 


8009 


8685 


8761 


8836 


76 


4 


8912 


8988 


9063 


9139 


9214 


9290 


9366 


9441 


9517 


9592 


76 


5 


9668 


9743 


9819 


9894 


9970 


760045 


760121 


760196 


760272 


760347 


75 


6 


760422 


760498 


760573 


760649 


760724 


0799 


0875 


0950 


1025 


1101 


75 


7 


1176 


1251 


1326 


1402 


1477 


1552 


1627 


1702 


1778 


1853 


75 


8 


1928 


2003 


2078 


2153 


2228 


2303 


2378 


2453 


2529 


2604 


75 


9 


2679 


2754 


2829 


2904 


2978 


3053 


3128 


3203 


3278 


3353 


75 


No. 


1 o 


1 1 


1 » 


1 3 


1 ± 


1 5 


1 6 


7 


* . 


9 


Djitf 



LOGARITHMS OF KUJ1BEHS. 



189 



No. | 



a I 3- | 4 j ft | 



8 | 9 jDiff. 



580 


763428 


763503 


763578 


763653 


763727 


763802 


763877 


763952 


764027 


764101 


75 


1 


4176 


4251 


4326 


4400 


4475 


4550 


4624 


4699 


4774 


4848 


75 


2 


4923 


4998 


5072 


5147 


5221 


5290 


5370 


5445 


5520 


5594 


75 


3 


5669 


5743 


5818 


5892 


5966 


6041 


6115 


6190 


6264 


6338 


74 


4 


6413 


6487 


6562 


6636 


6710 


6785 


6859 


6933 


7007 


7082 


74 


5 


7156 


7230 


7304 


7379 


7453 


7527 


7601 


7675 


7749 


7823 


74 


6 


7898 


7972 


8046 


8120 


8194 


8268 


8342 


8416 


8490 


8564 


74 


7 


8638 


8712 


8786 


8860 


8934 


9008 


9082 


9156 


9230 


9303 


74 


8 


9377 


9451 


9525 


9599 


9673 


9746 


9820 


9894 


9968 


770042 


74 


9 


770115 


770189 


770263 


770336 


770410 


770484 


770557 


770631 


770705 


0778 


74 


590 


770852 


770926 


770999 


771073 


771146 


771220 


771293 


771367 


771440 


771514 


74 


1 


1587 


1661 


1734 


1808 


1881 


1955 


2028 


2102 


2175 


2248 


73 


2 


2322 


2395 


2468 


2542 


2615 


2688 


2762 


2835 


2908 


2981 


73 


3 


3055 


3128 


3201 


3274 


3348 


3421 


3494 


'3567 


3640 


3713 


73 


4 


3786 


3860 


3933 


4006 


4079 


4152 


4225 


4298 


4371 


4444 


73 


5 


4517 


4590 


4663 


4736 


4809 


4882 


4955 


5028 


5100 


5173 


73 


6 


5246 


5319 


5392 


5465 


5538 


5610 


5683 


5756 


5829 


5902 


73 


7 


5974 


6047 


6J20 


6193 


6265 


6338 


6411 


6483 


6556 


6629 


73 


3 


6701 


6774 


6846 


6919 


6992 


7064 


7137 


7209 


7282 


7354 


73 


9 


7427 


7499 


7572 


7644 


7717 


7789 


7862 


7934 


8006 


8079 


72 


800 


778151 


778224 


778296 


778308 


778441 


778513 


778585 


778658 


778730 


778802 


72 


1 


8874 


8947 


9019 


9091 


9163 


9236 


9308 


9380 


9452 


9524 


72 


2 


9596 


9669 


9741 


9813 


9885 


9957 


780029 


780101 


780173 


780245 


72 


3 


780317 


780389 


780461 


780533 


780605 


780677 


0749 


0821 


0893 


0965 


72 


4 


1037 


1109 


1181 


1253 


1324 


1396 


1468 


1540 


1612 


1684 


72 


5 


1755 


1827 


1899 


1971 


2042 


2114 


2186 


2258 


2329 


2401 


72 


6 


2473 


2544 


2616 


2688 


2759 


2831 


2902 


2974 


3046 


3117 


72 


7 


3189 


3260 


3332 


3403 


3475 


3546 


3618 


3689 


3761 


3832 


71 


8 


3904 


3975 


4046 


4118 


4189 


4261 


4332 


4403 


4475 


4546 


71 


9 


4617 


4689 


4760 


4831 


4902 


4974 


5045 


5116 


5187 


5259 


71 


610 


785330 


785401 


785472 


785543 


785615 


785686 


785757 


785828 


785899 


785970 


71 


1 


6041 


6112 


6183 


6254 


6325 


6396 


6467 


6538 


6609 


6680 


71 


2 


6751 


6822 


6893 


6964 


7035 


7106 


7177 


7248 


7319 


7390 


71 


3 


7460 


7531 


7602 


7673 


7744 


7815 


7885 


7956 


8027 


8098 


71 


4 


8168 


8239 


8310 


8381 


8451 


8522 


8593 


8663 


8734 


8804 


71 


5 


8875 


8946 


9016 


9087 


9157 


9228 


9299 


9369 


9440 


9510 


71 


6 


9581 


9651 


9722 


9792 


98.63 


9933 


790004 


790074 


790144 


790215 


70 


7 


790285 


790356 


790426 


790496 


790567 


790637 


0707 


0778 


0848 


0918 


70 


8 


0988 


1059 


1129 


1199 


1269 


1340 


1410 


1480 


1550 


1620 


70 


9 


1691 


1761 


1831 


1901 


1971 


2041 


2111 


2181 


2252 


2322 


70 


620 


792392 


792462 


792532 


792602 


792672 


792742 


792812 


792882 


792952 


793022 


70 


1 


3092 


3162 


3231 


3301 


3371 


3441 


3511 


3581 


3651 


3721 


70 


2 


3790 


3860 


3930 


4000 


4070 


4139 


4239 


4279 


4349 


4418 


70 


3 


4488 


4558 


4627 


4697 


4767 


4836 


4906 


4976 


5045 


5115 


70 


4 


5185 


5254 


5324 


5393 


5463 


5532 


5602 


5672 


5741 


5811 


70 


5 


5880 


5949 


6019 


6088 


6158 


6227 


6297 


6366 


6436 


6505 


69 


6 


6574 


6644 


6713 


6782 


6852 


6921 


6990 


7060 


7129 


7198 


69 


7 


7268 


7337 


7406 


7475 


7545 


7614 


7683 


7752 


7821 


7890 


69 


8 


7960 


8029 


8098 


8167 


8236 


8305 


8374 


8443 


8513 


8582 


69 


9 


8651 


8720 


8789 


8858 


8927 


8996 


9065 


9134 


9203 


9272 


69 


530 


799341 


799409 


799478 


799547 


799616 


799685 


799754 


799823 


799892 


799961 


69 


1 


300029 


800098 


800167 


800236 


800305 


800373 


800442 


800511 


800580 


800648 


69 


2 


0717 


0786 


0854 


0923 


0992 


1061 


1129 


1198 


1266 


1335 


69 


3 


1404 


1472 


1541 


1609 


1678 


1747 


1815 


1884 


1952 


2021 


69 


4 


2089 


2158 


2226 


2295 


2363 


2432 


2500 


2568 


2637 


2705 


68 


5 


2774 


2842 


2910 


2979 


3047 


31J6 


3184 


3252 


3321 


3389 


68 


C 


3457 


3525 


3594 


3662 


3730 


3798 


3867 


3935 


4003 


4071 


68 


7 


4139 


4208 


4276 


4344 


4412 


4480 


4548 


4616 


4685 


4753 


68 


8 


4821 


4889 


4957 


5025 


5093 


5161 


5229 


5297 


5365 


5433 


68 


9 


5501 


5569 


5637 


5705 


5773 


5841 


5908 


5976 


6044 


6112 


68 



No. | 



2|3|4:|5|6|7j8|9|D<fi; 



190 



LOGARITHMS OF 2fU3fBERS. 



K<w | 


1 1 


1 3 


3 


1 * 


1 5 


1 6 


1 7 


8 


1 9 


I Dift 
68 


i>40 


806180 


806218 


806316 


S06384 |806451 


806519 


806587 


806655 


806723 


806790 


J 


6858 


6926 


6994 


7061 


7129 


7197 


7264 


7332 


7400 


7407[ 68 


2 


7535 


7603 


7670 


7738 


7806 


7873 


7941 


8008 


8076 


8143; 68 


3 


82 J 1 


8279 


8346 


8414 


8481 


8549 


8616 


8684 


8751 


8818 


67 


4 


8886 


8953 


9021 


9088 


9156 


9223 


9290 


9358 


9425 


9492 


67 


5 


9560 


9627 


9694 


9762 


9829 


9896 


9964 


810031 


810098 


810165 


67 


6 


8 J 0233 


810300 


810367 


810434 


810501 


810569 


810636 


0703 


0770 


0837 


67 


7 


0904 


0971 


1039 


1106 


1173 


1240 


1307 


1374 


1441 


1503 


67 


8 


1575 


1642 


1709 


1776 


1843 


1910 


1977 


2044 


2111 


2178 


67 


9 


2245 


2312 


2379 


2445 


2512 


2579 


2646 


2713 


2780 


2847 


67 


650 


8J2913 


812980 


813047 


813114 


813181 


813247 


813314 


813381 


813448 


813514 


67 


J 


3581 


3648 


3714 


3781 


3848 


3914 


3981 


4048 


4114 


4181 


67 


2 


4248 


4314 


4381 


4447 


4514 


4581 


4647 


4714 


4780 


4847 


67 


3 


4913 


4980 


5046 


5113 


5179 


5246 


5312 


5378 


5445 


5511 


66 


4 


5578 


5644 


5711 


5777 


5843 


5910 


5976 


6042 


6109 


6175 


66 


5 


6241 


6308 


6374 


6440 


6506 


6573 


6639 


6705 


6771 


6838 


66 


6 


6904 


6970 


7036 


7102 


7169 


7235 


7301 


7367 


7433 


7499 


66 


7 


7565 


7631 


7698 


7764 


7830 


7896 


7962 


8028 


8094 


8160 


66 


8 


8226 


8292 


8358 


8424 


8490 


8556 


8622 


8688 


8754 


8820 


66 


9 


8885 


8951 


9017 


9083 


9149 


9215 


9281 


9346 


9412 


9478 


66 


660 


819544 


819610 


819676 


819741 


819807 


819873 


819939 


820004 


820070 


820136 


66 


1 


820201 


820267 


820333 


820399 


820464 


820530 


820595 


0661 


0727 


0792 


66 


2 


0858 


0924 


0989 


1055 


1120 


1186 


1251 


1317 


1382 


1448 


66 


3 


1514 


1579 


1645 


1710 


1775 


1841 


1906 


1972 


2037 


2103 


C5 


4 


2168 


2233 


2299 


2364 


2430 


2495 


2560 


2626 


2691 


2750 


65 


5 


2822 


2887 


2952 


3018 


3083 


3148 


3213 


3279 


3344 


3409 


65 


6 


3474 


3539 


3605 


3670 


3735 


3800 


3865 


3930 


3990 


4061 


65 


7 


4126 


4191 


4256 


4321 


4386 


4451 


4516 


4581 


4646 


4711 


65 


8 


4776 


4841 


4906 


4971 


5036 


5101 


5166 


5231 


5296 


5361 


65 


9 


5426 


5491 


5556 


5621 


5686 


5751 


5815 


5880 


5945 


6010 


65 


670 


826075 


826140 


826204 


826269 


826334 


826399 


826464 


826528 


826593 


826658 


65 


1 


6723 


6787 


6852 


6917 


6981 


7046 


7111 


7175 


7240 


7305 


65 


2 


7369 


7434 


7499 


7563 


7628 


7692 


7757 


7821 


7886 


7951 


65 


3 


8015 


8080 


8144 


8209 


8273 


8338 


8402 


8467 


8531 


8595 


64 


4 


8660 


8724 


8789 


8853 


8918 


8982 


9046 


9111 


9175 


9239 


64 


5 


9304 


9368 


9132 


9497 


9561 


9625 


9690 


9754 


9818 


9882 


64 


6 


9947 


830011 


830075 


830139 


830204 


830268 


830332 


830396 


830460 


830525 


64 


7 


830589 


0653 


0717 


0781 


0845 


0909 


0973 


1037 


1102 


1166 


64 


8 


1230 


1294 


1358 


1422 


1486 


1550 


1614 


1678 


1742 


1806 


64 


9 


1870 


1934 


1998 


2062 


2126 


2189 


2253 


2317 


2381 


2445 


64 


680 


832509 


832573 


832637 


832700 


832764 


832828 


832892 


832956 


833020 


833083 


64 


i 


3147 


3211 


3275 


3338 


3402 


3466 


3530 


3593 


3657 


3721 


64 


2 


3784 3848 


3912 


3975 


4039 


4103 


4166 


4230 


4294 


4357 


64 


3 


4421 


4484 


4548 


4611 


4075 


4739 


4802 


4866 


4929 


4993 


64 


4 


5056 


5120 


5183 


5247 


5310 


5373 


5437 


5500 


5564 


5627 


63 


5 


5691 


5754 


5817 


5881 


5944 


6007 


6071 


6134 


6197 


6261 


63 


6 


6324 


6337 


6451 


6514 


6577 


6641 


6704 


6767 


6830 


6894 


63 


7 


6957 


7020 


7083 


7146 


7210 


7273 


7336 


7399 


7462 


7525 


03 


8 


7588 


7652 


7715 


7778 


7841 


7904 


7967 


8030 


8093 


8156 


63 


9 


8219 


8282 


8345 


8408 


8471 


8534 


8597 


8660 


8723 


8786 


63 


690 


838849 


338912 


838975 


839038 


839101 


839164 


839227 


839289 


839352 


839415 


63 


1 


9478 


9541 


9604 


9667 


9729 


9792 


9855 


9918 


9981 


840043 


63 


2 


8401)6 


840169 


840232 


840294 


840357 


840420 


840482 


840545 


840608 


067J 


63 


3 


0733 


0796 


0859 


0921 


0984 


1046 


1109 


1172 


1234 


12! )7 


63 


4 


1359 


1422 


1485 


1547 


1610 


1672 


1735 


1797 


1860 


1922 


63 


5 


1985 


2047 


2110 


2172 


2235 


2297 


2360 


2422 


2484 


2547 


02 


6 


2609 


2672 


2734 


2796 


2859 


2921 


2983 


3046 


3108 


3170 


02 


7 


3233 


3295 


3357 


3420 


3482 


3544 


3606 


3669 


3731 


3793 


62 


8 


38551 3918 


3980 


4042 


4104 


4166 


4229 


4291 


4353 


4415 


62 


9 


44771 4539 


4601 


4664 


4726 


4788 


4850 


4912 


4974 


5036 62 



No,|0|l,3|3|4:|5|6|7|8 



LOGARITHMS OF NUMBERS* 



191 



1 | H-3 | 4r | 5 | 6 | 7 | 8 | 9-|DJC 



700 


845093 


845160 


845222 


845284 


845346 


84.5408 


845470 


1845532 


845594 


845656 


62 


1 


5718 


5780 


5842 


5904 


5966 


6028 


6090 


6151 


6213 


6275 


62 


2 


6337 


6399 


6461 


6523 


6585 


6646 


6708 


6770 


6832 


6894 


62 


3 


6955 


7017 


7079 


7141 


7202 


7264 


7326 


7388 


7449 


7511 


62 


4 


7573 


7634 


7696 


7758 


7819 


7881 


7943 


8004 


8066 


8128 


62 


5 


8189 


8251 


8312 


8374 


8435 


8497 


8559 


8620 


8682 


8743 


62 


6 


8805 


8866 


8928 


8989 


9051 


9112 


9174 


9235 


9297 


9358 


61 


7 


9419 


9181 


9542 


9604 


9665 


9726 


9788 


9849 


9911 


9972 


61 


8 


850033 


850095 


85U156 


850217 


850279 


850340 


850401 


850462 


,850524 


850585 


6L 


9 


0646 


0707 


0769 


0830 


0891 


0952 


1014 


1075 


1136 


1197 


61 


710 


851258 


851320 


851381 


851442 


851503 


851564 


851625 


851686 


851747 


851809 


61 


1 


1870 


1931 


1992 


2053 


2114 


2175 


2236 


2297 


2358 


2419 


61 


2 


2480 


2541 


2602 


2663 


2724 


2785 


2846 


2907 


2968 


3029 


61 


3 


3090 


3150 


3211 


3272 


3333 


3394 


3455 


3516 


3577 


3637 


61 


4 


3698 


3759 


3820 


3881 


3941 


4002 


4063 


4124 


4185 


4245 


61 


5 


4306 


4367 


4428 


4488 


4549 


4610 


4670 


4731 


4792 


4852 


61 


6 


4913 


4974 


5034 


5095 


5156 


5216 


5277 


5337 


5398 


5459 


61 


7 


5519 


5580 


5640 


5701 


5761 


5822 


5882 


5943 


6003 


6064 


61 


8 


6124 


6185 


6245 


63C6 


6366 


6427 


6487 


6548 


6608 


6668 


60 


9 


6729 


6789 


6850 


6910 


6970 


7031 


7091 


7152 


7212 


7272 


60 


720 


857332 


857393 


857453 


857513 


857574 


857634 


837694 


857755 


857815 


857875 


60 


1 


7935 


7995 


8056 


8116 


8176 


8236 


8297 


8357 


8417 


8477 


60 


2 


8537 


8597 


8657 


8718 


8778 


8838 


8898 


8958 


9018 


9078 


60 


3 


9138 


9198 


9258 


9318 


9379 


9439 


9499 


9559 


9619 


9679 


60 


4 


9739 


9799 


9859 


9918 


9978 


860038 


860098 


860158 


860218 


860278 


60 


5 


860338 


860398 


860458 


860518 


860578 


0637 


0697 


0757 


0817 


0877 


60 


6 


0937 


0996 


1056 


1116 


1176 


1236 


1295 


1355 


1415 


1475 


60 


7 


1534 


1594 


1654 


1714 


1773 


1833 


1893 


1952 


2012 


2072 


60 


8 


2131 


2191 


2251 


2310 


2370 


2430 


2489 


2549 


2608 


2668 


60 


9 


2728 


2787 


2847 


2906 


2966 


3025 


3085 


3144 


3204 


3263 


60 


730 


863323 


863382 


863442 


863501 


863561 


863620 


863680 


863739 


863799 


863858 


59 


I 


3917 


3977 


4036 


4096 


4155 


4214 


4274 


4333 


4392 


4452 


59 


2 


4511 


4570 


4630 


4689 


4748 


4808 


4867 


4926 


4985 


5045 


59 


3 


5104 


5163 


5222 


5282 


5341 


5400 


5459 


5519 


5578 


5637 


59 


4 


5696 


5755 


5814 


5874 


5933 


5992 


6051 


6110 


6169 


6228 


59 


5 


6287 


6346 


6405 


6465 


6524 


6583 


6642 


6701 


6760 


6819 


59 


6 


6878 


6937 


6996 


7055 


7114 


7173 


7232 


7291 


7350 


7409 


59 


7 


7467 


7526 


7585 


7644 


7703 


7762 


7821 


7880 


7939 


7998 


59 


8 


8056 


8J15 


8174 


8233 


8292 


8350 


8409 


8468 


8527 


8586 


59 


9 


8644 


8703 


8762 


8821 


8879 


8938 


8997 


9056 


9114 


9173 


59 


740 


869232 


869290 


869349 


869408 


869466 


869525 


869584 


869642 


869701 


869760 


59 " 


1 


9818 


9877 


9935 


9994 


870053 


870111 


870170 


870228 


870287 


870345 


59 


2 


870404 


870462 


870521 


870579 


0638 


0696 


0755 


0813 


0872 


0930 


58 


3 


0989 


1047 


1106 


1164 


1223 


1281 


1339 


1398 


1456 


1515 


58 


4 


1573 


1631 


1690 


1748 


1806 


1865 


1923 


1981 


2040 


2098 


58 


5 


2156 


2215 


2273 


2331 


2389 


2448 


2506 


2564 


2622 


2681 


58 


6 


2739 


2797 


2855 


2913 


2972 


3030 


3088 


3146 


3204 


3262 


58 


7 


3321 


3379 


3437 


3495 


3553 


3611 


3669 


3727 


3785 


3844 


58 


8 


3902 


3960 


4018 


4076 


4134 


4192 


4250 


4308 


4366 


4424 


58 


9 


4482 


4540 


4598 


4656 


4714 


4772 


4830 


4888 


4945 


5003 


58 


750 


875061 


875119 


875177 


875235 


875293 


875351 


875409 


875466 


875524 


875582 


58 


1 


5640 


5698 


5756 


5813 


5871 


5929 


5987 


6045 


6102 


6160 


58 


2 


6218 


6276 


6333 


6391 


6449 


6507 


6564 


6622 


6680 


6737 


58 


3 


6795 


6853 


6910 


6968 


7026 


7083 


7141 


7199 


7256 


7314 


58 


4 


7371 


7429 


7487 


7544 


7602 


7659 


7717 


7774 


7832 


7889 


58 


5 


7947 


8004 


8062 


8119 


8177 


8234 


8292 


8349 


8407 


8464 


57 


6 


8522 


8579 


8637 


8694 


8752 


8809 


8866 


8924 


8981 


9039 


57 


7 


9096 


9153 


9211 


9268 


9325 


9383 


9440 


9497 


9555 


9612 


57 


8 


9669 


9726 


9784 


9841 


9898 


9956 


880013 


880070 


880127 


880185 


57 


9 


880242 


880299 


880356 


880413 


880471 


880528 


0585 


0642 


06991 0756) 


57 



Na|0|l|3|3|4|5|6|7|8|9 



iDiffr 



192 



LOGARITHMS OF NUMBERS. 



No.| | 1 | g | 3 | 4= | 5 | 6 | 7 | 8 \ 9 | Difi 



760 


880814 


880871 


880928 


880985 


881042 881099 


881156 881213 


881271 


881328 


57 


1 


1385 


1442 


1499 


1556 


1613 


1670 


1727 


1784 


1841 


1898 


57 


2 


1955 


2012 


2069 


2126 


2183 


2240 


2297 


2354 


2411 


2468 


57 


3 


2525 


2581 


2638 


2695 


2752 


2809 


2866 


2923 


2980 


3037 


57 


4 


3093 


3150 


3207 


3264 


3321 


3377 


3434 


3491 


3548 


3605 


51 


5 


3661 


3718 


3775 


3832 


3888 


3945 


4002 


4059 


4115 


4172 


57 


6 


4229 


4285 


4342 


4399 


4455 


4512 


4569 


4625 


4682 


4739 


57 


7 


4795 


4852 


4909 


4965 


5022 


5078 


5135 


5192 


5248 


5305 


57 


8 


5361 


5418 


5474 


5531 


5587 


5644 


5700 


5757 


5813 


5870 


57 


9 


5926 


5983 


6039 


6096 


6152 


6209 


6265 


6321 


6378 


6434 


56 


770 


886491 


886547 


886604 


886660 


886716 


886773 


886829 


886885 


886942 


886998 


56 


1 


7054 


7111 


7167 


7223 


7280 


7336 


7392 


7449 


7505 


7561 


56 


2 


7617 


7674 


7730 


7786 


7842 


7898 


7955 


8011 


8067 


8123 


56 


3 


8179 


8236 


8292 


8348 


8404 


8460 


8516 


8573 


8629 


8685 


56 


4 


8741 


8797 


8853 


8909 


8965 


9021 


9077 


9134 


9190 


9246 


56 


5 


9302 


9358 


9414 


9470 


9526 


9582 


9633 


9694 


9750 


9806 


56 


6 


9862 


9918 


9974 


890030 


890086 


890141 


890197 


890253 


890309 


890365 


56 


' 7 


890421 


890477 


890533 


0589 


0645 


0700 


0756 


0812 


0868 


0924 


56 


8 


0980 


1035 


1091 


1147 


1203 


1259 


1314 


1370 


1426 


1482 


56 


9 


1537 


1593 


1649 


1705 


1760 


1816 


1872 


1928 


1983 


2039 


56 


780 


892095 


892150 


892206 


892262 


892317 


892373 


892429 


892484 


892540 


892595 


56 


1 


2651 


2707 


2762 


2818 


2873 


2929 


2985 


3040 


3096 


3151 


56 


2 


3207 


3202 


3318 


3373 


3429 


3484 


3540 


3595 


3651 


3706 


56 


3 


3762 


3817 


3873 


'3928 


3984 


4039 


4094 


4150 


4205 


4261 


55 


4 


4316 


4371 


4427 


4482 


4538 


4593 


4648 


4704 


4759 


4814 


55 


5 


4870 


4925 


4980 


5036 


5091 


5146 


5201 


5257 


5312 


5367 


55 


6 


5423 


5478 


5533 


5588 


5644 


5699 


5754 


5809 


5864 


5920 


55 


7 


5975 


6030 


6085 


6140 


6195 


6251 


6306 


6361 


6416 


6471 


55 


8 


6526 


6581 


6636 


6692 


6747 


6802 


6857 


6912 


6967 


7022 


55 


9 


7077 


7132 


7187 


7242 


7297 


7352 


7407 


7462 


7517 


7572 


55 


790 


897627 


897682 


897737 


897792 


897847 


897902 


897957 


898012 


898067 


898122 


55 


1 


8176 


8231 


8286 


8341 


8396 


8451 


8506 


8561 


8615 


8670 


55 


2 


8725 


8780 


8835 


8890 


8944 


8999 


9054 


9109 


9164 


9218 


55 


3 


9273 


9328 


9383 


9437 


9492 


9547 


9602 


9656 


9711 


9766 


55 


4 


9821 


9375 


9930 


9985 


900039 


900094 


900149 


900203 


900258 


900312 


55 


5 


900367 


900422 


900476 


900531 


0586 


0640 


0695 


0749 


0804 


0859 


55 


6 


0913 


09G8 


1022 


1077 


1131 


1186 


1240 


1295 


1349 


1404 


55 


7 


1458 


1513 


1567 


1622 


1676 


1731 


1785 


1840 


1894 


1948 


54 


8 


2003 


2057 


2112 


2166 


2221 


2275 


2329 


2384 


2438 


2492 


54 


9 


2547 


2601 


2655 


2710 


2764 


2818 


2873 


2927 


2981 


3036 


54 


*800 


903090 


903144 


903199 


903253 


903307 


903361 


903416 


903470 


903524 


903578 


54 


1 


3633 


3687 


3741 


3795 


3849 


3904 


3958 


4012 


4066 


4120 


54 


2 


4174 


4229 


4283 


4337 


4391 


4445 


4499 


4553 


4607 


4661 


54 


3 


4716 


4770 


4824 


4878 


4932 


4986 


5040 


5094 


5148 


5202 


54 


4 


5256 


5310 


5364 


5418 


5472 


5526 


5580 


5634 


5688 


5742 


54 


5 


5796 


5850 


5904 


5958 


6012 


6066 


6119 


6173 


6227 


6281 


54 


6 


6335 


6389 


6443 


6497 


6551 


6604 


6658 


6712 


6766 


6820 


54 


7 


6874 


6927 


6981 


7035 


7089 


7143 


7196 


7250 


7304 


7358 


54 


e 


7411 


7465 


7519 


7573 


7026 


7680 


-7734 


7787 


7841 


7895 


54 


9 


7949 


8002 


8056 


8110 


8163 


8217 


8270 


8324 


8378 


8431 


54 


819 


908485 


908539 


908592 


908646 


908699 


908753 


908807 


908860 


908914 


908967 


54 


1 


9021 


9074 


9128 


9181 


9235 


9289 


9342 


9396 


9449 


9503 


54 


2 


9556 


9610 


9663 


9716 


9770 


9823 


9877 


9930 


9084 


910037 


53 


3 


910091 


910144 


910197 


910251 


910304 


910358 


910411 


910464 


910518 


0571 


53 


4 


0624 


0678 


0731 


0784 


0838 


0891 


0944 


0998 


1051 


1104 


53 


5 


1158 


1211 


1264 


1317 


1371 


1424 


1177 


1530 


1584 


1637 


53 


6 


1690 


1743 


1797 


1850 


1903 


1956 


2009 


2063 


2116 


2169 


53 


7 


2222 


2275 


2328 


2381 


2435 


2488 


2541 


2594 


2647 


2700 


53 


8 


2753 


2806 


2859 


2913 


2966 


3019 


3072 


3125 


3173 


3231 


53 


9 


3284 


3337 


3390 


3443 


3496 


3549 


3602 


3655 


3708 


3761 


53 



No. I | 1 | 3 ! 3 | 4 | 5 | 6 | 7 I 8 | 9 |Di* 



LOGARITHMS OF NUMBERS* 



193 



No. / 



1?3(314:|516|7|8|9. | DiflL 



820 
1 
2 
3 
4 
5 
6 
7 
8 
9 

830 
1 
2 
3 
4 
5 
6 
7 
8 
9 

840 
1 

2 
3 
4 
5 
6 
7 
8 
9 

850 
1 
2 
3 
4 
5 
6 
7 
8 
9 

860 
1 
2 
3 
4 
5 
6 
7 
8 
9 

70 
1 
2 
3 
4 
5 
6 
7 
8 
9 



913814 
4343 

4872 
5400 
5927 
6454 
6980 
7506 
8030 
8555 

919078 
960] 

920123 
0645 
1166 
1686 
2206 
2725 
3244 
3762 

924279 
4796 
5312 

5828 
6342 
6857 
7370 
7883 
8396 
8908 

929419 
9930 

930440 
0949 
1458 
1966 
2474 
2981 
3487 
3993 

934498 
5003 
5507 
6011 
6514 
7016 
7518 
8019 
8520 
9020 

939519 
940018 
0516 
3014 
1511 
2008 
2504 
3000 
3495 



913867 
4396 
4925 
5453 
5980 
6507 
7033 
7558 
8083 
8607 

919130 
9653 

920176 
0697 
1218 
1738 
2258 
2777 
3296 
3814 

924331 

4848 
5364 
5879 
6394 
6908 
7422 
7935 
8447 
8959 

929470 
9981 

930491 
1000 
1509 
2017 
2524 
3031 
3538 
4044 

934549 
5054 
5558 
6061 
6564 
7066 
7568 
8069 
8570 
9070 

939569 
940068 
0566 
1064 
1561 
2058 
2554 
3049 
3544 
4038 



913920 
4449 
4977 
5505 
6033 
6559 
7085 
7611 
8135 
8659 

919183 
9706 

920228 
0749 
1270 
1790 
2310 
2829 
3348 
3865 

924383 
4899 
5415 
5931 
6445 
6959 
7473 
7986 
8498 
9010 

929521 
930032 
0542 
1051 
1560 
2068 
2575 
3082 
3589 
4094 

934599 
5104 
5608 
6111 
6614 
7117 
7618 
8119 
3620 
9120 

t)39619 
940118 
0616 
1114 
1611 
2107 
2603 
3099 
3593 
4088 



913973 
4502 
5030 
5558 
6085 
6612 
7138 
7663 
8188 
8712 

919235 
9758 

920280 
0801 
1322 
1842 
2362 
2881 
3399 
3917 

924434 
4951 
5467 
5982 
6497 
7011 
7524 
8037 
8549 
9061 

929572 
930083 
0592 
1102 
1610 
2118 
2626 
3133 
3639 
4145 

934650 
5154 
5658 
6162 
6665 
7167 
7668 
8169 
8670 
9170 

939669 
940168 
0666 
1163 
1660 
2157 
2653 
3148 
3643 
4137 



914026 
4555 
5083 
5611 
6138 
6664 
7190 
7716 
8240 
8764 

919287 
9810 

920332 
0853 
1374 
1894 
2414 
2933 
3451 
3969 

924486 
5003 
5518 
6034 
6548 
7062 
7576 



9112 

1929623 
930134 
0643 
1153 
1661 
2169 
2677 
3183 
3690 
4195 

934700 
5205 
5709 
6212 
6715 
7217 
7718 
8219 
8720 
9220 

939719 
940218 
07161 
1213 
1710 
2207 
2702' 
3198 
3692 1 
4:86 i 



914079 
4608 
5136 
5664 
6191 
6717 
7243 
' 7768 
8293 
8816 

919340 
9862 

920384 
0906 
1426 
1946 
2466 
2985 
3503 
4021 

924538 
5054 
5570 
6085 
6600 
7114 
7627 
8140 
8652 
9163 

929674 
930185 
0694 
1204 
1712 
2220 
2727 
- 3234 
3740 
4246 

934751 
5255 
5759 
6262 
6765 
7267 
7769 
8269 
8770 
9270 

939769 
940267 
0765 
1263 
1760 
2256 
2752 
3247 
3742 
4236 



914132 
4660 
5189 
5716 
6243 
6770 
7295 
7820 
8345 
8869 

919392 
9914 

920436 
0958 
1478 
1998 
2516 
3037 
3555 
4072 

924589 
5106 
5621 
6137 
6651 
7165 
7678 
8191 
8703 
9215 

929725 
930236 
0745 
1254 
1763 
2271 
2778 
3285 
3791 
4296 

934801 
5306 
5809 
6313 
6815 
7317 
7819 
8320 
8820 
9320 

939819 

940317 

0815 

1313 

1809 
2306 
2801 
3297 
3791 
4285 



914184 


914237 


914290 


4713 


4766 


4819 


5241 


5294 


5347 


5769 


5822 


5875 


6296 


6349 


6401 


6822 


6875 


6927 


7348 


7400 


7453 


7873 


7925 


7978 


8397 


8450 


8502 


8921 


8973 


9026 


919444 


919496 


919549 


9967 


920019 


920071 


920489 


0541 


0593 


1010 


1062 


1114 


1530 


1582 


1634 


2050 


2102 


2154 


2570 


2622 


2674 


3089 


3140 


3192 


3607 


3658 


3710 


4124 


4176 


4228 


924641 


924693 


924744 


5157 


5209 


5261 


5673 


5725 


5776 


6188 


6240 


6291 


6702 


6754 


6805 


7216 


7268 


7319 


7730 


7781 


7832 


8242 


8293 


8345 


8754 


8805 


8857 


9266 


9317 


9368 


929776 


929827 


929879 


930287 


930338 


930389 


0796 


0847 


0898 


1305 


1356 


1407 


1814 


1865 


1915 


2322 


2372 


2423 


2829 


2879 


2930 


3335 


3386 


3437 


3841 


3892 


3943 


4347 


4397 


4448 


934852 


934902 


934953 


5356 


5406 


5457 


5860 


5910 


5960 


6363 


6413 


6463 


6865 


6916 


6966 


7367 


7418 


7468 


7869 


7919 


7969 


8370 


8420 


8470 


8870 


8920 


8970 


9369 


9419 


9469 


939869 


939918 


939968 


940367 


940417 


940467 


0865 


0915 


0964 


1362 


1412 


1462 


1859 


1909 


1958 


2355 


2405 


2455 


2851 


2901 


2950 


3346 


3396 


3445 


3841 


3890 


3939 


4335 


4384 


4433 



2 | 3 



4: J 5 | 6 | 7 | 8 | 9 !Di4 



194 



LOGARITHMS OF NUMBERS. 



3 | 4 



5 | 6 | 7 | 8 | 9 |Di« 



880 


944483 


944532 


944581 


944631 


944680 


944729 


944779 


944828 


944877 


944927 


49 


1 


4976 


5025 


5074 


5124 


5173 


5222 


5272 


5321 


5370 


5419 


49 


2 


5469 


5518 


5567 


5616 


5665 


5715 


5764 


5813 


5862 


5912 


49 


3 


5961 


6010 


6059 


6108 


6157 


6207 


6256 


6305 


6354 


6403 


49 


4 


6452 


6501 


6551 


6600 


6649 


6698 


6747 


6796 


6845 


6894 


4£ 


5 


6943 


6992 


7041 


7090 


7140 


7189 


7238 


7287 


7336 


7385 


49 


6 


7434 


7483 


7532 


7581 


7630 


7679 


7728 


7777 


7826 


7875 


49 


7 


7924 


7973 


8022 


8070 


8119 


8168 


8217 


8266 


8315 


8364 


49 


8 


8413 


8462 


8511 


8560 


8609 


8657 


8706 


8755 


8804 


8853 


49 


9 


8902 


8951 


8999 


9048 


9097 


9146 


9195 


9244 


9292 


9341 


49 


890 


949390 


949439 


949488 


949536 


949585 


949634 


949683 


949731 


949780 


949829 


49 


1 


9878 


9926 


9975 


950024 


950073 


950121 


950170 


950219 


950267 


950316 


49 


2 


950365 


950414 


950462 


0511 


0560 


0608 


0657 


0706 


0754 


0803 


49 


3 0851 


0900 


094i> 


0997 


1046 


1095 


1143 


1192 


1240 


1289 


49 


4 


1338 


1386 


1435 


1483 


1532 


1580 


1629 


1677 


1726 


1775 


49 


5 


1823 


1872 


1920 


1969 


2017 


2066 


2114 


2163 


2211 


2260 


48 


6 


2308 


2356 


2405 


2453 


2502 


2550 


2599 


2647 


2696 


2744 


48 


7 


2792 


2841 


2889 


2938 


2986 


3034 


3083 


3131 


3180 


3228 


48 


8 


3276 


3325 


3373 


3421 


3470 


3518 


3566 


3615 


3663 


3711 


48 


9 


3760 


3808 


3856 


3905 


3953 


4001 


4049 


4098 


4146 


4194 


48 


900 


954243 


954291 


954339 


954387 


954435 


954484 


954532 


954580 


954628 


954677 


48 


1 


4725 


4773 


4821 


4869 


4918 


4966 


5014 


5062 


5110 


5158 


48 


2 


5207 


5255 


5303 


5351 


5399 


5447 


5495 


5543 


5592 


5640 


48 


3 


5688 


5736 


5784 


5832 


5880 


5928 


5976 


6024 


6072 


6120 


48 


4 


6168 


6216 


6265 


6313 


6361 


6409 


6457 


6505 


6553 


6601 


48 


5 


6649 


6697 


6745 


6793 


6840 


6888 


6936 


6984 


7032 


7080 


48 


6 


7128 


7176 


7224 


7272 


7320 


7368 


7416 


7464 


7512 


7559 


48 


7 


7607 


7655 


7703 


7751 


7799 


7847 


7894 


7942 


7990 


8038 


48 


8 


8086 


8134 


8181 


8229 


8277 


8325 


8373 


8421 


8468 


8516 


48 


9 


8564 


8612 


8659 


8707 


8755 


8803 


8850 


8898 


8946 


8994 


48 


910 


959041 


959089 


959137 


959185 


959232 


959280 


959328 


959375 


959423 


959471 


48 


1 


9518 


9566 


9614 


9661 


9709 


9757 


9804 


9852 


9900 


9947 


48 


2 


9995 


960042 


960090 


960138 


960185 


960233 


960281 


960328 


960376 


960423 


48 


3 


960471 


0518 


0566 


0613 


0661 


0709 


0756 


0804 


0851 


0899 


48 


4 


0946 


0994 


1041 


1089 


1136 


1184 


1231 


1279 


1326 


1374 


48 


5 


1421 


1469 


1516 


1563 


1611 


1658 


1706 


1753 


1801 


1848 


47 


6 


1895 


1943 


1990 


2038 


2085 


2132 


2180 


2227 


2275 


2322 


47 


7 


2369 


2417 


2464 


2511 


2559 


2606 


2653 


2701 


2748 


2795 


47 


8 


2843 


2890 


2937 


2985 


3032 


3079 


3126 


3174 


3221 


3268 


47 


9 


3316 


3363 


3410 


3457 


3504 


3552 


3599 


3646 


3693 


3741 


47 


920 


963788 


963835 


963882 


963929 


963977 


964024 


964071 


964118 


964165 


964212 


47 


1 


4260 


4307 


4354 


4401 


4448 


4495 


4542 


4590 


4637 


4684 


47 


2 


4731 


4778 


4825 


4872 


4919 


4966 


5013 


5061 


5108 


5155 


47 


3 


5202 


5249 


5296 


5343 


5390 


5437 


5484 


5531 


5578 


5625 


47 


4 


5672 


5719 


5766 


5813 


5860 


5907 


5954 


6001 


6048 


6095 


47 


5 


6142 


6189 


6236 


6283 


6329 


6376 


6423 


6470 


6517 


6564 


47 


6 


6611 


6658 


6705 


6752 


6799 


6845 


6892 


6939 


6986 


7033 


47 


7 


7080 


7127 


7173 


7220 


7267 


7314 


7361 


7408 


7454 


7501 


47 


8 


7548 


7595 


7642 


7688 


7735 


7782 


7829 


7875 


7922 


7969 


47 


9 


8016 


8062 


8109 


8156 


8203 


8249 


8296 


8343 


8390 


8436 


47 


930 


968483 


968530 


968576 


968623 


968670 


968716 


968763 


968810 


968856 


968903 


47 


1 


8950 


8996 


9043 


9090 


9136 


9183 


9229 


9276 


9323 


9369 


47 


2 


9416 


9463 


9509 


9556 


9602 


9649 


9695 


9742 


9789 


9835 


47 


3 


9882 


9928 


9975 


970021 


970068 


970114 


970161 


970207 


970254 


970300 


47 


4 


970347 


970393 


970440 


0486 


0533 


0579 


0626 


0672 


0719 


0765 


46 


5 


0812 


0858 


0904 


0951 


0997 


1044 


1090 


1137 


1183 


1229 


46 


6 


1276 


1322 


1369 


1415 


1461 


1508 


1554 


1601 


1647 


1693 


46 


7 


1740 


1786 


1832 


1879 


1925 


1971 


2018 


2064 


2110 


2157 


46 


8 


2203 


2249 


2295 


2342 


2388 


2434 


2481 


2527 


2573 


2619 


46 


9 


2666 


2712 


2758 


2804 


2851 


2897 


2943 


2989 


3035 


3082 


46 



No.| O | 1 | » |3J4|5|6|7|8|!>! Dift 



LOGARITHMS OF NUMBERS. 



195 



No. 


o 


1 


1 3 


1 3 


1 ± 


5 


6 


7 


8 


9 


1 Otfl 


940 


973128 


973174 


973220 


973266 


973313 


973359 


973405 


973451 


973497 


973543 


46 


J 


3590 


3636 


3682 


3728 


3774 


3820 


3866 


3913 


3959 


4005 


46 


2 


4051 


4097 


4143 


4189 


4235 


4281 


4327 


4374 


4420 


4466 


46 


3 


4512 


4558 


4604 


4650 


4696 


4742 


4788 


4834 


4880 


4926 


46 


4 


4972 


5018 


5064 


5110 


515G 


5202 


5248 


5294 


5340 


5386 


46 


5 


5432 


5478 


5524 


5570 


5616 


5662 


5707 


5753 


5799 


5845 


46 


6 


5891 


5937 


5983 


6029 


6075 


6121 


6167 


6212 


6258 


6304 


46 


7 


6350 


6396 


6442 


6488 


6533 


6579 


6625 


6671 


6717 


6763 


46 


8 


6808 


6854 


6900 


6946 


6992 


7037 


7083 


7129 


7175 


7220 


46 


9 


7266 


7312 


7358 


7403 


7449 


7495 


7541 


7586 


7632 


7678 


46 


950 


977724 


977769 


977815 


977861 


977906 


977952 


977998 


978043 


978089 


978135 


46 


J 


8181 


8226 


8272 


8317 


8363 


8409 


8454 


8500 


8546 


8591 


46 


o 


8637 


8683 


8728 


8774 


8819 


8865 


8911 


8956 


9002 


9047 


46 


3 


9093 


9138 


9184 


9230 


9275 


9321 


9366 


9412 


9457 


9503 


46 


4 


9548 


9594 


9639 


9685 


9730 


9776 


9821 


9867 


9912 


9958 


46 


5 


980003 


980049 


980094 


980140 


980185 


980231 


980276 


980322 


980367 


980412 


45 


6 


0458 


0503 


0549 


0594 


0640 


0685 


0730 


0776 


0821 


0867 


46 


7 


0912 


0957 


1003 


1048 


1093 


1139 


1184 


1229 


1275 


1320 


45 


8 


1366 


1411 


1456 


1501 


1547 


1592 


1637 


1683 


1728 


1773 


45 


9 


1819 


1864 


1909 


1954 


2000 


2045 


2090 


2135 


2181 


2226 


45 


860 


982271 


982316 


982362 


982407 


982452 


982497 


982543 


982588 


982633 


982678 


45 


1 


2723 


2769 


2814 


2859 


2904 


2949 


2994 


3040 


3085 


3130 


45 


2 


3175 


3220 


3265 


3310 


3356 


3401 


3446 


3491 


3536 


3581 


45 


3 


3626 


3671 


3716 


3762 


3807 


3852 


3897 


3942 


3987 


4032 


45 


4 


4077 


4122 


4167 


4212 


4257 


4302 


4347 


4392 


4437 


4482 


45 


5 


4527 


4572 


4617 


4662 


4707 


4752 


4797 


4842 


4887 


4932 


45 


6 


4977 


5022 


5067 


5112 


5157 


5202 


5247 


5292 


5337 


5382 


45 


7 


5426 


5471 


5516 


5561 


5606 


5651 


5696 


5741 


5786 


5830 


45 


8 


5875 


5920 


5965 


6010 


6055 


6100 


6144 


6189 


6234 


6279 


45 


9 


6324 


6369 


6413 


6458 


6503 


6548 


6593 


6637 


6682 


6727 


45 


970 


986772 


986817 


986861 


986906 


986951 


986996 


987040 


987085 


987130 


987175 


45 


1 


7219 


7264 


7309 


7353 


7398 


7443 


7488 


7532 


7577 


7622 


45 


2 


7666 


7711 


7756 


7800 


7845 


7890 


7934 


7979 


8024 


8068 


45 


3 


8113 


8157 


8202 


8247 


8291 


8336 


8381 


8425 


8470 


8514 


45 


4 


8559 


8604 


8648 


8693 


8737 


8782 


8826 


8871 


8916 


8960 


45 


5 


9005 


9049 


9094 


9138 


9183 


9227 


9272 


9316 


9361 


9405 


45 


6 


9450 


9494 


9539 


9583 


9628 


9672 


9717 


9761 


9806 


9850 


44 


7 


9895 


9939 


9983 


990028 


990072 


990117 


990161 


990206 


990250 


990294 


44 


8 


990339 


990383 


990428 


0472 


0516 


0561 


0605 


0650 


0694 


0738 


44 


9 


0783 


0827 


0871 


0916 


0960 


1004 


1049 


1093 


1137 


1182 


44 


980 


991226 


991270 


991315 


991359 


991403 


991448 


991492 


991536 


991580 


991625 


44 


1 


1669 


1713 


1758 


1802 


1846 


1890 


1935 


1979 


2023 


2067 


44 


2 


2111 


2156 


2200 


2244 


2288 


2333 


2377 


2421 


2465 


2509 


44 


3 


2554 


2598 


2642 


2686 


2730 


2774 


2819 


2863 


2907 


2951 


44 


4 


2995 


3039 


3083 


3127 


3172 


3216 


3260 


3304 


3348 


3392 


44 


5 


3436 


3480 


3524 


3568 


3613 


3657 


3701 


3745 


3789 


3833 


44 


6 


3877 


3921 


3965 


4009 


4053 


4097 


4141 


4185 


4229 


4273 


44 


7 


4317 


4361 


4405 


4449 


4493 


4537 


4581 


4625 


4669 


4713 


44 


8, 4757 


4801 


4845 


4889 


4933 


4977 


5021 


5065 


5108 


5152 


44 


9 


5196 


5240 


5284 


5328 


5372 


5416 


5460 


5504 


5547 


5591 


44 


990 


995635 


995679 


995723 


995767 


995811 


995854 


995898 


995942 


995986 


996030 


44 


1 


6074 


6117 


6161 


6205 


6249 


6293 


6337 


6380 


6424 


6468 


44 


2 


6512 


6555 


6599 


6643 


6687 


6731 


6774 


6818 


6862 


6900 


44 


3 


6949 


6993 


7037 


7080 


7124 


7168 


7212 


7255 


7299 


7343 


44 


4 


7386 


7430 


7474 


7517 


7561 


7605 


7648 


7692 


7736 


7779 


44 


5 


7823 


7867 


7910 


7954 


7998 


8041 


8085 


8129 


8172 


8216 


44 


6 


8259 


8303 


8347 


8390 


8434 


8477 


8521 


8564 


8608 


8652 


44 


7 


8695 


8739 


8782 


8826 


8869 


8913 


8956 


9000 


9043 


9087 


44 


8 


9131 


9174 


9218 


9261 


9305 


9348 


9392 


9435 


9479 


9522 


44 


9 


9565 


9609 


9652 


9696 


9739 


9783 


9826 


9870 


9913 


9957 


43 



Wo. I O j 1 I % I 3 



5 \ 6 | 7 j 8 I 9 \m 



TABLE XIII. 

LOGARITHMIC SINES, COSINES, TANGENTS, AND 
COTANGENTS. 



N. B. — The minutes in the left-hand column of each page, 
Increasing downwards, belong to the degrees at the top ; and 
those increasing upwards, in the right-hand column, belong to 
the degrees below. 

In using the differences for one second, in columns D, the 
two right-hand figures should be marked off as decimals. 
Thus the difference for log. sin. 1° 12' 5" would be 99.82 
X 5 = 499.1, additive to the mantissa .321027 treated as an 
integer, and the difference for log. cos. 8° 30' 50" would be 
0.30 X 50 = 16.0 subtractive from the mantissa .995203 treated 
as an integer. 

The differences in columns D range opposite the upper one 
of the two functions to which they respectively apply. 

The first column D refers to Sines, the second to Cosines, 
tb<> third to both Tangents and Cotangents. 



198 CO Degree.) LOGARITHMIC ZINES, COSWES> ETC. 



M. 


| Sine 


! D. 


1 Cosine 


I D. 


! Tang. 


1 D. 


Cotang-. 





Inf. Negr. 




10-000000 




0-000000 


1 


Infinite. 


1 


6463726 


501717 


000000 


00 


6-463726 


1 501717 


13536274 


2 


764756 


293485 


000000 


00 


764756 


! 293485 


235244 


3 


940847 


208231 


000000 


00 


940847 


208231 


059153 


4 


7-065786 


161517 


000000 


00 


7-065786 


161517 


12*934214 


5 


162696 


131968 


000000 


00 


162696 


131969 


837304 


6 


241877 


11 1578 


9-999999 


01 


241878 


111578 


758122 


7 


308824 


96653 


999999 


01 


308825 


99653 


691175 


8 


366816 


85254 


999999 


01 


366817 


85254 


633183 


9 


417968 


76263 


999999 


01 


417970 


76263 


582030 


10 


463725 


68988 


999998 


01 


463727 


68988 


536273 


11 


7*505118 


62981 


9-999998 


01 


7-505120 


62981 


12-494880 


12 


542906 


57936 


999997 


01 


542909 


57938 


457091 


13 


577668 


53641 


999997 


01 


577672 


53642 


422328 


14 


609853 


49938 


999996 


01 


609857 


49939 


390143 


15 


639816 


46714 


999996 


01 


639820 


46715 


360180 


16 


667845 


43881 


999995 


01 


667849 


43882 


332151 


17 


694173 


41372 


999995 


01 


694179 


41373 


305821 


18 


718997 


39135 


999994 


01 


719003 


39136 


280997 


19 


742477 


37127 


999993 


01 


742484 


37128 


257516 


20 


764754 


35315 


999993 


01 


764761 


35317 


235239 


21 


7-785943 


33672 


9-999992 


01 


7-785951 


33673 


12-214049 


22 


806146 


32175 


999991 


01 


806155 


32176 


193845 


23 


825451 


30805 


999990 


01 


825460 


30806 


174540 


24 


843934 


29547 


999989 


02 


843944 


29549 


156056 


25 


861662 


28388 


999988 


02 


861674 


28390 


138326 


26 


878695 


27317 


999988 


02 


878708 


27318 


121292 


27 


895085 


26323 


999987 


02 


895099 


26325 


104901 


28 


910879 


25399 


999986 


02 


910894 


25401 


089106 


29 


926119 


24538 


999985 


02 


926134 


24540 


073866 


30 


940842 


23733 


999983 


02 


940858 


23735 


059142 


31 


7-955082 


22980 


9-999982 


02 


7-955100 


22981 


12044900 


32 


968870 


22273 


999981 


02 


968889 


22275 


031 111 


33 


982233 


21608 


999980 


02 


982253 


21610 


017747 


34 


995198 


20981 


999979 


02 


995219 


20983 


004781 


35 


8-007787 


20390 


999977 


02 


8-007809 


20392 


11-992191 


36 


020021 


19831 


999976 


02 


020045 


19833 


979955 


37 


031919 


19302 


999975 


02 


031945 


19305 


968055 


38 


043501 


18801 


999973 


02 


043527 


18803 


956473 


39 


054781 


18325 


999972 


02 


054809 


18327 


945191 


40 


065776 


17872 


999971 


02 


065806 


17874 


934194 


41 


8076500 


17441 


9-999969 


02 


8-076531 


17444 


11-923469 


42 


086965 


17031 


999968 


02 


086997 


17034 


913003 


43 


097183 


16639 


999966 


02 


097217 


16642 


902783 


44 


107167 


16265 


999964 


03 


107202 


16268 


892797 


45 


116926 


15908 


999963 


03 


116903 


15910 


883037 


46 


126471 


15566 


999961 


03 


126510 


15568 


873490 


47 


135810 


15238 


999959 


0^ 


135851 


15241 


864149 


48 


144953 


14924 


999958 


03 


144996 


14927 


855004 


49 


153907 


14622 


999956 


03 


153952 


14627 


846048 


50 


162681 


14333 


999954 


03 


162727 


14336 


837273 


51 


8171280 


14054 


9-999952 


03 


8171328 


14057 


11-828672 


52 


179713 


13786 


999950 


03 


179763 


13790 


820237 


53 


187985 


13529 


999948 


03 


188036 


13532 


811964 


54 


196102 


13280 


999946 


03 


196156 


13284 


803844 


55 


204070 


13041 


999944 


03 


204126 


13044 


795874 


56 


211895 


12810 


999942 


04 


211953 


12814 


788047 


57 


219581 


12587 


999940 


04 


219641 


12590 


780359 


58 


227134 


12372 


999938 


04 


227195 


12376 


772805 


59 


234557 


12164 


999936 


04 


234621 


12168 


765379 


60 


241855 


11963 


999934 


04 


241921 


11967 


758079 



| Coeiue | 



Sine | | Cotang. 

89 Degrees. 



) Tang. | M. 



LOGARITHMIC SINES, COSINES, ETC. (1 Degree.) 199 



M. ' 



Sine 



1 Cosine | D. | 



8241855 
249033 
256094 
263042 



276614 
233243 

289773 
296207 
302546 
308794 

8-314954 
321027 
327016 
332924 
338753 
344504 
350181 
355783 
3613i5 
366777 

8-372171 
377499 
382762 
387962 
393101 
398179 
403199 
408161 
413068 
417919 

8-422717 
427462 
432156 
436800 
441394 
445941 
450440 
454893 
459301 
463665 

8-467985 
472263 
476498 
480693 
484848 
488963 
493040 
497078 
501080 
505045 

8-508974 
512867 
516726 
520551 
524343 
528102 
531828 
535523 
539186 
542819 



11963 
11768 
11580 
11398 
11221 
11050 
10883 
10721 
10565 
10413 
10266 
10122 
9982 
9847 
9714 
9586 
9460 
9338 
9219 
9103 
8990 
8880 
8772 
8667 
8564 
8464 
8366 
8271 
8177 
8086 
7996 
7909 
7823 
7740 
7657 
7577 
7499 
7422 
7346 
7273 
7200 
7129 
7060 
6991 
6924 
6859 
6794 
6731 
6669 
6608 
6548 
6489 
6431 
6375 
6319 
6264 
6211 
6158 
6106 
6055 
6004 



9-999934 
999932 
999929 
999927 
999925 
999922 
999920 
999918 
999915 
9999 i 3 
999910 

9-999907 
999905 
999902 
999899 
999897 
999894 
999891 
999888 
999885 
999882 

9-999879 
999876 
999873 
999870 
999867 
999864 
999861 
999858 
999854 
999851 

9-999848 
999844 
999841 
999838 
999834 
999831 
999827 
999823 
999820 
999816 

9-999812 
999809 
999805 
999801 
999797 
999793 
999790 
999786 
999782 
999778 

9999774 
999769 
999765 
999761 
999757 
99 753 
999748 
999744 
999740 
999735 



04 
04 
04 
04 
04 
04 
04 
04 
04 
04 
04 
04 
04 
04 
05 
05 
05 
05 
05 
05 
05 
05 
05 
05 
05 
05 
05 
05 
05 
05 
06 
06 
06 
06 
06 
06 
06 
06 
06 
06 
06 
06 
06 
06 
06 
07 
07 
07 
07 
07 
07 
07 
07 
07 
07 
07 
07 
07 
07 
07 
07 



Tang. 


D. | 


Cotan£. 




8-241921 


11967 


11-758079 


60 


249102 


11772 


750898 


59 


256165 


11584 


743835 


58 


2631 15 


11402 


736885 


57 


269956 


11225 


730044 


56 


276691 


11054 


723309 


55 


283323 


10887 


716677 


54 


289856 


10726 


710J44 


53 


296292 


10570 


703708 


52 


302634 


10418 


697366 


51 


308884 


10270 


691116 


50 


8-315046 


10126 


11-684954 


49 


321122 


9987 


678878 


48 


3271 14 


9851 


672886 


47 


333025 


9719 


666975 


46 


338856 


9590 


661144 


45 


344610 


9465 


655390 
64 7; 11 


44 


350289 


9343 


43 


355895 


9224 


644105 


42 


361430 


9108 


638570 


41 


366895 


8995 


633105 


40 


8372292 


8885 


11-627708 


39 


377622 


8777 


622378 


38 


382889 


8672 


617111 


37 


388092 


8570 


611908 


36 


393234 


8470 


606766 


35 


398315 


8371 


601685 


34 


403338 


8276 


596662 


33 


408304 


8182 


591096 


32 


413213 


8091 


586787 


bl 


418068 


8002 


581932 


30 


8-422869 


7914 


11-577131 


29 


427618 


7830 


572382 


•28 


432315 


7745 


567685 


27 


436962 


7663 


563038 


20 


441560 


7583 


558440 


25 


446110 


7505 


553890 


24 


450613 


7428 


549387 


23 


455070 


7352 


544930 


22 


459481 


7279 


540519 


-2*1 


463849 


7206 


536151 


20 


8-468172 


7135 


11-531828 


19 


472454 


7066 


527546 


18 


476693 


6998 


523307 


17 


480892 


6931 


519108 


16 


485050 


6865 


514950 


15 


489170 


6801 


510830 


14 


493-250 


6738 


506750 


13 


497293 


6676 


502707 


12 


501298 


6615 


498702 


11 


505267 


6555 


494733 


ifl 


8-509200 


6496 


11490800 


9 


513098 


6439 


486902 


8 


516961 


6382 


483039 


7 


520790 


6326 


479210 


6 


524586 


6272 


475414 


5 


528349 


6218 


471651 


4 


532080 


6165 


467920 


3 


535779 


6113 


464221 


2 


539447 


6062 


460553 


1 


543084 


6012 


456916 


(1 



| Cosine | 



1 Sine 



| Cotang. 1 



I Tang. | M, 



88 Degrees. 



200 (2 Degrees.) LOGARITHMIC SINES, COSINES, ETC. 



M. | 


Sine 


D. 


Cosine 


D. 


Tang. 


D. 


Cotang. 


1 





8-542819 


6004 


9-999735 


07 


8543084 


6012 


11456916 


60 


1 


546422 


5955 


999731 


07 


546691 


5962 


453309 


59 


2 


549995 


5906 


999726 


07 


550268 


5914 


449732 


58 


3 


553539 


5858 


999722 


08 


553817 


5866 


446183 


57 


4 


557054 


5811 


999717 


08 


557336 


5819 


442664 


56 


5 


560540 


5765 


999713 


08 


560828 


5773 


439172 


55 


6 


563999 


5719 


999708 


08 


564291 


5727 


435709 


54 


7 


567431 


5674 


999704 


08 


567727 


5682 


432273 


53 


8 


570836 


5630 


999699 


08 


571137 


5638 


4288G3 


52 


9 


574214 


5587 


999694 


08 


574520 


5595 


425480 


51 


JO 


577566 


5544 


999689 


08 


577877 


5552 


422123 


50 


11 


8-580892 


5S02 


9-999685 


08 


8-581208 


5510 


11-418792 


49 


12 


584193 


5460 


999680 


08 


584514 


5408 


415486 


48 


13 


587469 


5419 


999675 


08 


587795 


5427 


412205 


47 


U 


590721 


5379 


999670 


08 


591051 


5387 


408949 


46 


15 


593948 


5339 


999665 


08 


594283 


5347 


405717 


45 


1G 


597152 


5300 


999660 


08 


597492 


5308 


402508 


44 


17 


600332 


5261 


999655 


08 


600677 


5270 


399323 


43 


18 


603489 


5223 


999650 


08 


603839 


5232 


396161 


42 


19 


606623 


5186 


999645 


09 


606978 


5194 


393022 


41 


20 


609734 


5149 


999640 


09 


610094 


5158 


389906 


40 


21 


8612823 


5112 


9-999635 


09 


8-613189 


5121 


11-386811 


39 


22 


615891 


5076 


999629 


09 


616262 


5085 


383738 


38 


23 


618937 


5041 


999624 


09 


619313 


5050 


380687 


37 


24 


621962 


5006 


999619 


09 


622343 


5015 


377657 


36 


25 


624965 


4972 


999614 


09 


625352 


4981 


374648 


35 


26 


627948 


4938 


999608 


09 


628340 


4947 


371600 


34 


27 


630911 


4904 


999603 


09 


631308 


4913 


308692 


33 


28 


633854 


4871 


999597 


09 


634256 


4880 


365744 


32 


29 


636776 


4839 


999592 


09 


637184 


4848 


362816 


31 


30 


639680 


4806 


999586 


09 


640093 


4816 


359907 


30 


31 


8 642563 


4775 


9-999581 


09 


8-642982 


4784 


11-357018 


29 


32 


645428 


4743 


999575 


09 


645853 


4753 


354147 


28 


33 


648274 


4712 


999570 


09 


648704 


4722 


351296 


27 


34 


651102 


4682 


9995G4 


09 


651537 


4691 


348403 


20 


35 


653911 


4652 


999558 


10 


654352 


4661 


345648 


25 


36 


656702 


4622 


999553 


10 


657149 


4631 


342851 


24 


37 


659475 


4592 


999547 


10 


659928 


4602 


340072 


23 


38 


662230 


4563 


999541 


10 


662689 


4573 


337311 


22 


39 


664968 


4535 


999535 


10 


665433 


4544 


334507 


21 


40 


667689 


4506 


999529 


10 


668160 


4526 


331840 


20 


41 


8670393 


4479 


9-999524 


10 


8-670870 


4488 


11-329130 


19 


42 


673080 


4451 


999518 


10 


673563 


4461 


326437 


18 


43 


675751 


4424 


999512 


10 


676239 


4434 


323761 


17 


44 


678405 


4397 


999506 


10 


678900 


4417 


321100 


16 


45 


681043 


4370 


999500 


10 


681544 


4380 


318456 


15 


46 


683665 


4344 


999493 


10 


684172 


4354 


315828 


14 


47 


686272 


4318 


999487 


10 


686784 


4328 


313216 


13 


48 


688863 


4292 


999481 


10 


689381 


4303 


310619 


12 


49 


691438 


4267 


999475 


10 


691 903 


4277 


308037 


11 


50 


693998 


4242 


999469 


10 


694529 


4252 


305471 


10 


51 


8-696543 


4217 


9-909463 


11 


8-697081 


4228 


11-302919 


9 


52 


699073 


4192 


999456 


11 


699617 


4203 


300383 


8 


53 


701589 


4168 


999450 


11 


702139 


4179 


297801 


7 


54 


704090 


4144 


999443 


11 


704646 


4155 


295354 


6 


55 


706577 


4121 


999437 


11 


707140 


4132 


292860 


5 


56 


709049 


4097 


999431 


11 


709618 


4108 


290382 


4 


57 


711507 


4074 


999424 


11 


712083 


4085 


287917 


3 


58 


713952 


4051 


999418 


11 


714534 


4062 


285465 


2 


59 


716383 


4029 


999411 


11 


716972 


4040 


283028 


1 


60 ' 


718800 


4006 


999404 


11 


719396 


4017 


280604 






I _ Cosine | ( Sine I \ Cotang. | 

87 Degrees. 



I Tang. I M. 





LOGARITHMIC SIXES, 


COSIXES, ETC. (3 Degrees.) 5 


>01 


M | 


Sine | 


D. 1 


Cosine I 


D. | 


Tang. | 


D. I 


Cotang. | 




o 


8-718800 


4006 


9-999404 


11 


8-719396 


4017 


U -280604 1 


60 


1 


721204 


3984 


999398 


11 


721806 


3995 


278194 


59 


2 


723595 


3962 


999391 


11 


724204 


3974 


275796 


58 


3 


725972 


3941 


999384 


11 


726588 


3952 


273412 


57 


4 


728337 


3919 


999378 


11 


728959 


3930 


271041 


56 


5 


730688 


3898 


999371 


11 


731317 


3909 


268683 


55- 


6 


733027 


3877 


999364 


12 


733663 


3889 


266337 


54 


7 


735354 


3857 


999357 


12 


735996 


3868 


264004 


53 


8 


737667 


3836 


999350 


12 


738317 


3848 


261683 


52 


9 


739959 


3816 


999343 


12 


740626 


3827 


259374 


51 


10 


742259 


3796 


999336 


12 


742922 


3807 


257078 


50 


jl 


8-744536 


3776 


9-999329 


12 


8-745207 


3787 


11.254793 


49' 


12 


746802 


3756 


999322 


12 


747479 


3768 


252521 


48 


13 


749055 


3737 


999315 


12 


749740 


3749 


250260 


47 


14 


751297 


3717 


999308 


12 


751989 


3729 


2480 i : 


46 




753528 


3698 


999301 


12 


754227 


3710 


245773 


45 


16 


755747 


3679 


999294 


12 


756453 


3692 


243547 


44 


17 


757955 


3661 


999286 


12 


758668 


3673 


241332 


43 


18 


760151 


3642 


999279 


12 


760872 


3655 


239128 


42 


19 


762337 


3624 


999272 


12 


763065 


3636 


236935 


41 


20 


764511 


3606 


999265 


12 


765246 


3618 


234754 


40 


21 


8-766675 


3588 


9-999257 


12 


8-71.7417 


3600 


11-232583 


39 


22 


768828 


3570 


999250 


13 


769578 


3583 


230422 


38 


23 


770970 


3553 


999242 


13 


771727 


3565 


228273 


37 


24 


773i01 


3535 


999235 


13 


773866 


3548 


226134 


36 


25 


775223 


3518 


999227 


13 


775995 


3531 


224005 


35 


26 


777333 


3501 


999220 


13 


778114 


3514 


221886 


34 


27 


779434 


3484 


999212 


13 


780222 


3497 


219778 


33 


28' 


781524 


3467 


999205 


13 


782320 


3480 


217680 


32 


29 


783605 


3451 


999197 


13 


784408 


3464 


215592 


31 


30 


785675 


3431 


999189 


13 


786486 


3447 


213514 


30 


31 


8-787736 


3418 


9-999181 


13 


8-788554 


3431 


11-211446 


29 


32 


789787 


3402 


999174 


13 


790613 


M14 


209387 


28^ 


33 


791828 


3386 


999166 


13 


792662 


3399 


207338 


27 


34 


793859 


3370 


999158 


13 


794701 


3383 


205299 


26 


35 


795881 


3354 


999150 


13 


796731 


3368 


203269 


25 


36 


797894 


3339 


999142 


13 


798752 


3352 


201248 


24 


37 


799897 


3323 


999134 


^3 


800763 


3337 


199237 


23 


38 


801892 


3308 


999126 


13 


802765 


3322 


197235 


22 


39 


803876 


3293 


999118 


13 


804758 


3307 


195242 


21 


40 


805852 


3278 


999110 


13 


806742 


3292 


193258 


20 


41 


8-807819 


3263 


9-999102 


13 


8-808717 


3278 


11191283 


19 


42 


809777 


3249 


999094 


14 


810683 


3262 


189317 


18 


43 


811726 


3234 


999086 


14 


812641 


3248 


187359 


17 


44 


813667 


3219 


999077 


14 


814589 


3233 


185411 


16 


45 


815599 


3205 


999069 


14 


816529 


3219 


183471 


15 


46 


817522 


3191 


999061 


14 


818461 


3205 


181539 


14 


47 


819436 


3177 


999053 


14 


820384 


3191 


179616 


13 


48 


821343 


3163 


999044 


14 


822298 


3177 


177702 


12 


49 


823240 


3149 


999036 


14 


824205 


3163 


175795 


11 


50 


825130 


3135 


999027 


14 


826103 


3150 


173897 


10 


51 


8-827011 


3122 


9-999019 


14 


8-827992 


3136 


11172008 


9 


52 


828884 


3108 


999010 


14 


829874 


3123 


170126 


8 


53 


830749 


3095 


999002 


14 


831748 


3110 


168252 


7 


54 


832007 


3082 


998993 


14 


833613 


3096 


166387 


6 


55 


834456 


3069 


998984 


14 


835471 


3083 


164529 


5 


56 


836297 


3056 


998976 


14 


837321 


3070 


162679 


4 


57 


838130 


3043 


998967 


15 


839163 


3057 


160837 


3 


58 


839956 


3030 


998958 


15 


840998 


3045 


159002 


2 


59 


841774 


3017 


998950 


15 


842825 


3032 


157175 


1 


60 


843585 
J Cosine 


3000 


998941 


15 


1 844644 


3019 


155356 







i 


) Sue 


A 


j Cotang. 


1 


I Tang. 


I M. 








8 


3 Degr 


3es. 









202 (4 Degrees.) LOGARITHMIC SINES, COSINES, ETC. 



M. 


| Sine 


1 D. 


Cosine 


1 u. 


1 Tans. 


I D. 


Cotang. 







8-843585 


3005 


9-998941 


15 


8-844644 


3019 


11-155356 


60 


1 


845387 


2992 


998932 


15 


846455 


3007 


153545 


59 


2 


847183 


2980 


998923 


15 


848260 


2995 


151740 


58 


3 


848971 


2967 


998914 


15 


850057 


2982 


149943 


57 


4 


850751 


2955 


998905 


15 


851846 


2970 


148154 


56 


5 


852525 


2943 


998896 


15 


853628 


2958 


146372 


55 


6 


854291 


2931 


998887 


15 


855403 


2946 


144597 


54 


7 


856049 


2919 


998878 


15 


857171 


2935 


142829 


53 


8 


857801 


2907 


998869 


15 


858932 


2923 


141068 


52 


9 


859546 


2896 


998860 


15 


860686 


2911 


139314 


51 


10 


861283 


2884 


998851 


15 


862433 


2900 


137567 


50 


11 


8-863014 


2873 


9-998841 


15 


8-864173 


2888 


11 135827 


49 


12 


864738 


2861 


998832 


15 


665906 


2877 


134094 


48 


13 


866455 


2850 


" 998823 


16 


867032 


2866 


132368 


47 


14 


868165 


2839 


998813 


16 


869351 


2854 


130649 


46 


15 


869868 


2828 


998804 


16 


871064 


2843 


128936 


45 


16 


871565 


2817 


998795 


16 


872770 


2832 


127230 


44 


17 


873255 


2806 


998785 


16 


874469 


2821 


125531 


43 


18 


874938 


2795 


998776 


16 


876162 


2811 


123838 


42 


19 


876615 


2786 


998766 


16 


877849 


2800 


122151 


41 


20 


878285 


2773 


998757 


16 


879529 


2789 


120471 


40 


21 


8-879949 


2763 


9-998747 


16 


8-881202 


2779 


11-118798 


39 


22 


881607 


2752 


998738 


16 


882859 


2768 


117131 


38 


23 


883258 


2742 


998728 


16 


884530 


2758 


115470 


37 


24 


884903 


2731 


998718 


16 


886185 


2747 


113815 


36 


23 


886542 


2721 


998708 


16 


887833 


2737 


1121G7 


35 


23 


888174 


2711 


998699 


16 


• 889476 


2727 


110524 


34 


27 


889801 


2700 


998689 


16 


891112 


2717 


108888 


33 


28 


891421 


2690 


998679 


16 


892742 


2707 


107258 


32 


29 


893035 


2680 


998669 


17 


894306 


2697 


105634 


31 


30 


894643 


2670 


998659 


17 


895984 


2687 


104016 


30 


31 


8-896246 


2660 


9-998649 


17 


8-897596 


2677 


11-102404 


29 


32 


897842 


2051 


998639 


17 


899203 


2667 


100797 


28 


33 


899432 


2641 


998629 


17 


900803 


2658 


099197 


27 


34 


901017 


2631 


998619 


17 


902398 


2648 


097002 


26 


35 


902596 


2622 


998609 


17 


903987 


2638 


096013 


25 


36 


904169 


2612 


998599 


17 


905570 


2629 


094430 


24 


37 


905736 


2603 


998589 


17 


907147 


2620 


092853 


23 


38 


907297 


2593 


998578 


17 


908719 


2610 


091281 


22 


39 


908853 


2584 


998568 


17 


910285 


2601 


089715 


21 


40 


910404 


2575 


998558 


17 


911846 


2592 


088154 


20 


41 


8-911949 


2566 


9-998548 


17 


8-913401 


2583 


11-086599 


19 


42 


913488 


2556 


998537 


17 


914951 


2574 


085049 


18 


43 


915022 


2547 


998527 


17 


910495 


2565 


083505 


17 


44 


916550 


2538 


998516 


lO 


918034 


2556 


081966 


16 


45 


918073 


2529 


998506 


18 


919568 


2547 


080432 


15 


46 


919591 


2520 


998495 


18 


921096 


2538 


078904 


14 


47 


921103 


2512 


998485 


18 


922619 


2530 


077381 


13 


48 


922610 


2503 


998474 


18 


924136 


2521 


075864 


12 


49 


924112 


2494 


998464 


18 


925649 


2512 


074351 


11 


50 


925609 


2486 


998453 


18 


927156 


2503 


072844 


10 


51 


8-927100 


2477 


9998442 


18 


8-928658 


2495 


11-071342 


9 


52 


928587 


2469 


998431 


18 


930155 


2486 


069845 


8 


53 


930068 


2460 


998421 


18 


931647 


2478 


068353 


7 


54 


931544 


2452 


998410 


18 


933134 


2470 


066866 


6 


55 


933015 


2443 


998399 


18 


934616 


2461 


065384 


5 


56 


934481 


2435 


998388 


18 


936093 


2453 


063907 


4 


57 


935942 


2427 


998377 


18 


937565 


2445 


062435 


3 


58 


937398 


2419 


998366 


18 


939032 


2437 


0609(58 


2 


59 


938850 


2411 


998355 


18 


940494 


2430 


059506 


1 


60 


940296 


2403 


998344 


18 


941952 


2421 


058048 






| Cosine \ 



Sine | | Cotang. 

85 Degrees, 



I Tang. | M, 



LOGARITHMIC SINES, COSINES, ETC. (5 Degrees.) 203 



Sine 



I D. | 



8-940296 
94 J 738 
943174 
944606 
946034 
947456 
948874 
930287 
951696 
953100 
954499 

8-955894 
957284 
958670 
960052 
961429 
962801 
964170 
965534 
966893 
968249 

8969600 
970947 
972289 
973628 
974962 
976293 
977619 
978941 
980259 
981573 

8-982883 
984189 
985491 
986789 



989374 
990660 
991943 
993222 
994497 

8-995768 
997036 
998299 
999560 

9-000816 
002069 
003318 
004563 
005805 
007044 

9^008278 
009510 
010737 
011962 
013182 
014400 
015613 
016824 

59 I 018031 

60 ' 019235 



2403 
2394 
2387 
2379 
2371 
2363 
2355 
2348 
2340 
2332 
2325 
2317 
2310 
2302 
2295 
2288 
2280 
2273 
2266 
2259 
2252 
2244 
2238 
2231 
2224 
2217 
2210 
2203 
2197 
2i90 
2183 

2177 

2170 
2163 
2157 
2150 
2144 
2138 
2131 
2125 
2119 
2112 
2106 
2100 
2094 
2087 
2082 
2076 
2070 
2064 
2058 
2052 
2046 
2040 
2034 
2029 
2023 
2017 
2012 
2006 
2000 



Cosine 


1 D. 


Tang. 


1 o. 


r Cotanff. 




9-99834*. 


19 


8-941952 


2421 


11058048 


60 


998333 


19 


943404 


2413 


056596 


59 


998322 


19 


944852 


2405 


055148 


58 


99831 1 


19 


946295 


2397 


053705 


57 


998300 


19 


947734 


2390 


052266 


56 


998289 


19 


949168 


2382 


050832 


55 


998277 


19 


950597 


2374 


049403 


54 


998266 


19 


952021 


2366 


047979 


53 


998255 


19 


953441 


2360 


046559 


52 


998243 


19 


954856 


2351 


045144 


51 


998232 


19 


956267 


2344 


043733 


53 


9-998220 


19 


8-957674 


2337 


11 042326 


49 


998209 


19 


959075 


2329 


040925 


48 


998197 


19 


960473 


2323 


039527 


47 


998186 


19 


961*66 


2314 


038134 


46 


998174 


19 


963255 


2307 


036745 


45 


998163 


19 


964639 


2300 


035361 


44 


998151 


19 


966019 


2293 


033981 


43 


998139 


20 


967394 


2286 


032606 


42 


998128 


20 


968766 


2279 


031234 


41 


998116 


20 


970133 


2271 


029867 


40 


9-998104 


20 


8-971496 


2265 


11028504 


39 


998092 


20 


972855 


2257 


027145 


38 


998080 


20 


974209 


2251 


025791 


37 


998068 


20 


975560 


2244 


024440 


36 


998056 


20 


976906 


2237 


023094 


35 


998044 


20 


978248 


2230 


021752 


34 


998032 


20 


979586 


2223 


020414 


33 


998020 


20 


980921 


2217 


019079 


32 


998008 


20 


982251 


2210 


017749 


31 


997996 


20 


983577 


2204 


016423 


30 


9-997984 


20 


8-984899 


2197 


11015101 


29 


997972 


20 


986217 


2191 


013783 


28 


997959 


20 


987532 


2184 


012468 


27 


997947 


20 


988842 


2178 


011158 


26 


997935 


21 


990149 


2171 


009851 


25 


997922 


21 


991451 


2165 


008549 


24 


997910 


21 


992750 


2158 


007250 


23 


997897 


21 


994045 


2152 


005955 


22 


997885 ■ 


21 


995337 


2146 


004663 


21 


997872 


21 


996624 


2140 


003376 


20 


9-997860 


21 


8-997908 


2134 


11-002092 


19 


997847 


21 


999188 


2127 


000812 


18 


997835 


21 


9-000465 


2121 


10-999535 


17 


997822 


21 


001738 


2115 


998262 


16 


997809 


21 


003007 


2109 


996993 


15 


997797 


21 


004272 


2103 


995728 


H 


997784 


21 


005534 


2097 


994466 


13 


997771 


21 


006792 


2091 


993208 


12 


997758 


21 


008047 


2085 


991953 


11 


997745 


21 


009298 


2080 


990702 


10 


9-997732 


21 


9 010546 


2074 


10-989454 


9 


997719 


21 


011790 


2068 


988210 


8 


997706 


21 


013031 


2062 


986969 


7 


997693 


22 


014268 


2056 


985732 


6 


997680 


22 


015502 


2051 


984498 


5 


997667 


22 


016732 


2045 


983268 


4 


997654 


22 


017959 


2040 


982041 


3 


997641 


22 


019183 


2033 


980817 


a 


997628 


22 


020403 


2028 


979597 


i 


997614 


22 


021620 


2023 


978380 






| Cosine | 



Sine ' | | Cotang. 

84 Degrees. 



I Tang. | M. 



204 (6 Degrees.) LOGARITHMIC SINES, COSINES, ETC. 



M. 


| Sine 


1 D. 


| Cosine 


1 D. 


| Tang. 


1 D. 


Cotang. 


1 





9019235 


2000 


9-997614 


22 


9021620 


2023 


10-978380 


60 


1 


020435 


1995 


997601 


22 


022834 


2017 


977166 


59 


2 


021632 


1989 


997588 


22 


024044 


2011 


975956 


58 


3 


022825 


1984 


997574 


22 


025251 


2006 


974749 


57 


4 


024016 


1978 


997561 


22 


026455 


2000 


973545 


56 


5 


025203 


1973 


997547 


■ 22 


027655 


1995 


972345 


55 


6 


026386 


1967 


997534 


23 


028852 


1990 


971148 


54 


7 


027567 


1962 


997520 


23 


030046 


1985 


969954 


53 


8 


028744 


1957 


997507 


23 


031237 


1979 


968763 


52 


9 


029918 


1951 


997493 


23 


032425 


1974 


967575 


51 


10 


031089 


1947 


997480 


23 


033609 


1969 


966391 


50 


n 


9032257 


1941 


9-997466 


23 


9-034791 


1964 


10-965209 


49 


12 


033421 


1936 


997452 


23 


035969 


1958 


964031 


48 


13 


034582 


1930 


997439 


23 


037144 


1953 


962856 


47 


14 


035741 


1925 


997425 


23 


038316 


1948 


961684 


46 


15 


036896 


1920 


997411 


23 


039485 


1943 


960515 


45 


16 


038048 


1915 


997397 


23 


040651 


1938 


959349 


44 


17 


039197 


1910 


997383 


23 


041813 


1933 


958187 


43 


18 


040342 


1905 


997369 


23 


042973 


1928 


957027 


42 


19 


041485 


1899 


997355 


23 


044130 


1923 


955870 


41 


20 


042625 


1894 


997341 


23 


045284 


1918 


954716 


40 


21 


9 043762 


1889 


9-997327 


24 


9-046434 


1913 


10-953566 


39 


22 


044895 


1884 


997313 


24 


047582 


1908 


952418 


38 


23 


046026 


1879 


997299 


24 


048727 


1903 


951273 


37 


24 


047154 


1875 


997285 


24 


049869 


1898 


950131 


36 


25 


048279 


1870 


997271 


24 


051008 


1893 


948992 


35 


26 


049400 


1865 


997257 


24 


052144 


1889 


947856 


34 


27 


050519 


1860 


997242 


24 


053277 


1884 


946723 


33 


28 


051635 


1855 


997228 


24 


054407 


1879 


945593 


32 


29 


052749 


1850 


997214 


24 


055535 


1874 


944465 


31 


30 


053859 


1845 


997199 


24 


056659 


1870 


943341 


30 


31 


054966 


1841 


9-997185 


24 


9-057781 


1865 


10942219 


29 


32 


056071 


1836 


997170 


24 


058900 


1869 


941100 


28 


33 


057172 


1831 


997156 


24 


060016 


1855 


939984 


27 


34 


058271 


1827 


997141 


24 


061130 


1851 


938870 


26 


35 


059367 


1822 


997127 


24 


062240 


1846 


937760 


25 


36 


060460 


1817 


997112 


24 


063348 


1842 


936652 


24 


37 


061551 


1813 


997098 


24 


064453 


1837 


935547 


23 


38 


062639 


1808 


997083 


25 


065556 


1833 


934444 


22 


39 


063724 


1804 


997068 


25 


066655 


1828 


833345 


21 


40 


064806 


1799 


997053 


25 


067752 


1824 


932248 


20 


41 


9-065885 


1794 


9997039 


25 


9-068846 


1819 


10931154 


19 


42 


066962 


1790 


997024 


25 


069938 


1815 


930062 


18 


43 


068036 


1786 


997009 


25 


071027 


1810 


928973 


17 


44 


069107 


1781 


996994 


25 


072113 


1806 


927887 


16 


45 


070176 


1777 


996979 


25 


073197 


1802 


926803 


15 


46 


071242 


1772 


996964 


25 


074278 


1797 


925722 


14 


47 


072306 


1768 


996949 


25 


075356 


1793 


924644 


13 


48 


073366 


1763 


996934 


25 


076432 


1789 


923568 


12 


49 


074424 


1759 


996919 


25 


077505 


1784 


922495 


11 


50 


075480 


1755 


996904 


25 


078576 


1780 


921424 


10 


51 


9076533 


1750 


9-996889 


25 


9-079644 


1776 


10920356 


9 


52 


077583 


1746 


996874 


25 


080710 


1772 


919290 


8 


53 


078631 


1742 


996858 


25 


081773 


1767 


918227 


7 


54 


079676 


1738 


996843 


25 


082833 


1763 


917167 


6 


55 


080719 


1733 


996828 


25 


083891 


1759 


916109 


5 


56 


081759 


1729 


996812 


26 


084947 


1755 


915053 


4 


57 


082797 


1725 


996797 


26 


086000 


1751 


914000 


3 


58 


083832 


1721 


996782 


26 


087050 


1747 


912950 


2 


59 


084864 


1717 


996766 


26 


088098 


1743 


911902 


1 


60 


085894 


1713 


996751 


26 


089144 


1738 


910856 






* Cowne I 



| Sine | 



| Cotang. | 



I Tang. I M. 



83 Degrees. 



LOGARITHMIC SINES, COSINES, ETC. (7 Degrees.) 2 05 



M. 


) Sine 


1 a 


| Cosine 


1 D. 


I Tanor. 


1 D. 


| Cotang 


1 





9*085894 


1713 


9*996751 


26 


9-089144 


1738 


10-910856 


60 


1 


086922 


1709 


996)735 


26 


090187 


1734 


909813 


59 


2 


087947 


1704 


996720 


26 


091228 


1730 


908772 


58 


3 


088970 


1700 


996)704 


26 


092266 


1727 


907734 


57 


4 


0890.90 


](*)6 


996688 


26 


093302 


1722 


906698 


56 


5 


091008 


1692 


996073 


26 


094336 


1719 


905664 


55 


6 


092024 


1688 


996657 


26 


095367 


1715 


904633 


54 


7 


09*037 


1684 


99664 1 


26 


096395 


1711 


903605 


53 


8 


094047 


1680 


9961)25 


36 


097422 


1707 


902578 


52 


9 


095056 


1676 


996610. 


26 


098446 


1703 


901554 


51 


JO 


096062 


1673 


996594 


26 


0994(58 


1699 


900532 


50 


11 


9-097065 


1668 


9-996578 


27 


9-100487 


1695 


10-899513 


49 


12 


098066 


1665 


996562 


27 


101504 


1691 


898496 


48 


13 


099065 


1661 


996546 


27 


102519 


1687 


897481 


47 


14 


100062 


1657 


990530 


27 


103532 


1684 


896468 


46 


15 


101056 


1653 


996514 


27 


104542 


1680 


895458 


45 


16 


1020-48 


1649 


996498 


27 


105550 


1676 


894450 


44 


11 


103037 


1645 


996482 


27 


106556 


1672 


893444 


43 


18 


104025 


1641 


996465 


27 


107559 


1669 


89244 1 


42 


19 


105010 


1638 


996449 


27 


108560 


1665 


891440 


41 


20 


105992 


1634 


996433 


27 


109559 


1661 


890441 


40 


21 


9-106973 


1630 


9-996417 


27 


9110556 


1658 


10-889444 


39 


22 


107951 


1627 


996400 


27 


111551 


1654 


888449 


38 


23 


108927 


1623 


996384 


27 


1 12543 


1650 


887457 


37 


24 


109901 


1619 


996368 


27 


113533 


1646 


886467 


36 


25 


110873 


1616 


996351 


27 


114521 


1643 


885479 


35 


26 


11 1842 


1612 


996335 


27 


115507 


1639 


884493 


34 


27 


112809 


1608 


996318 


27 


116491 


1636 


883509 


33 


28 


113774 


1605 


996302 


28 


117472 


1632 


882528 


32 


29 


114737 


1601 


996285 


28 


118452 


1629 


881548 


31 


30 


115698 


1597 


996269 


28 


119429 


1625 


880571 


30 


31 


9 116656 


1594 


9996252 


28 


9-120404 


1622 


10-879596 


29 


32 


117613 


1590 


996235 


28 


121377 


1618 


878623 


28 


33 


118567 


1587 


996219 


28 


122348 


1615 


877652 


27 


34 


119519 


1583 


996202 


28 


123317 


1611 


876683 


26 


35 


120469 


1580 


996185 


28 


124284 


1607 


875716 


25 


36 


121417 


1576 


996168 


28 


125249 


1604 


874751 


24 


37 


122362 


1573 


996151 


—28 


126211 


1601 


873789 


23 


38 


123306 


1569 


996134 


28 


127172 


1597 


872828 


22 


39 


124248 


1566 


996117 


28 


128130 


1594 


871870 


21 


40 


125187 


1562 


996100 


28 


129087 


1591 


870913 


20 


41 


9126125 


1559 


9-996083 


29 


9130041 


1587 


10-869959 


19 


42 


127060 


1556 


996066 


29 


130994 


1584 


869006 


18 


43 


127993 


1552 


996049 


29 


131944 


1581 


868056 


17 


44 


128925 


1549 


996032 


29 


132893 


1577 


867107 


16 


45 


129854 


1545 


996015 


29 


133839 


1574 


866161 


15 


16 


130781 


1542 


995998 


29 


134784 


1571 


865216 


14 


47 


131706 


1539 


995980 


29 


135726 


1567 


864274 


13 


48 


132630 


1535 


995963 


29 


136667 


1564 


863333 


12 


49 


133551 


1532 


995946 


29 


137605 


1561 


862395 


11 


50 


134470 


1529 


995928 


29 


138542 


1558 


861458 


10 


51 


9135387 


1525 


9-995911 


29 


9139476 


1555 


10-860524 


9 


52 


136303 


1522 


995894 


29 


140409 


1551 


859591 


8 


53 


137216 


1519 


995876 


29 


141340 


1548 


858660 


7 


54 


138128 


1516 


995859 


29 


142269 


1545 


857731 


6 


55 


139037 


1512 


995841 


29 


143196 


1542 


856804 


5 


50 


139944 


1509 


995823 


29 


144121 


1539 


855879 


4 


57 


140850 


1506 


995806 


29 


145044 


1535 


854956 


3 


58 


141754 


1503 


995788 


29 


145966 


1532 


854034 


2 


59 


142655 


1500 


995771 


29 


146885 


1529 


853115 


1 


SO 


143555 


1496 


995753 


29 


147803 


1526 


852197 






Cotine J 



I Cotang. | 



| Tang. 1 M. 



$4 .Degrees. 



206 (8 Degrees.) LOGARITHMIC SIXES, COS WES, ETC, 



M. 


| Sine 


1 D. 


| Cosine 


1 D. 


Tan*. 


1 D. 


Cotang. 


1 





9143555 


1496 


9-995753 


30 


9147803 


1526 


10.852197 


60 


1 


144453 


1493 


995735 


30 


148718 


1523 


851282 


59 


2 


145349 


1490 


995717 


30 


149632 


1520 


850368 


58 


3 


146243 


1487 


995699 


30 


150544 


1517 


849456 


57 


4 


147136 


1484 


995681 


30 


J 5 1454 


1514 


848546 


56 


5 


148026 


1481 


995664 


30 


152363 


1511 


847637 


55 


6 


148915 


1478 


995646 


30 


153209 


1508 


846731 


54 


7 


149802 


1475 


995628 


30 


154174 


1505 


845826 


53 


8 


150686 


1472 


995610 


30 


155077 


1502 


844923 


52 


9 


151569 


1469 


995591 


30 


155978 


1499 


844022 


51 


10 


152451 


1466 


995573 


30 


156877 


1496 


843123 


50 


11 


9153330 


1463 


9-995555 


30 


9-157775 


1493 


10-842225 


49 


12 


154208 


1460 


995537 


30 


158671 


1490 


841329 


48 


13 


155083 


1457 


995519 


30 


159565 


1487 


840435 


47 


14 


155957 


1454 


995501 


31 


160457 


1484 


839543 


46 


15 


156830 


1451 


995482 


31 


161347 


1481 


838653 


45 


16 


157700 


1448 


995464 


31 


162236 


1479 


837764 


44 


17 


158569 


1445 


995446 


31 


163123 


1476 


836877 


43 


J8 


159435 


1442 


995427 


31 


164008 


1473 


835992 


42 


19 


160301 


1439 


995409 


31 


164892 


1470 


835108 


41 


20 


161164 


1436 


995390 


31 


165774 


1467 


834226 


40 


21 


9-162025 


1433 


9-995372 


31 


9166654 


1464 


10833346 


39 


22 


162885 


1430 


995353 


31 


167532 


1461 


832468 


38 


23 


163743 


1427 


995334 


31 


108409 


1458 


831591 


37 


24 


164600 


1424 


995316 


31 


169284 


1455 


830716 


36 


25 


165454 


1422 


995297 


31 


170157 


1453 


829843 


35 


26 


166307 


1419 


995278 


31 


171029 


1450 


828971 


34 


27 


167159 


1416 


995260 


31 


171899 


1447 


828101 


3? 


28 


168008 


1413 


995241 


32 


172767 


1444 


827233 


35 


29 


168856 


1410 


995222 


32 


173634 


1442 


826366 


31 


30 


169702 


1407 


995203 


32 


174499 


1439 


825501 


30 


31 


9-170547 


1405 


9-995184 


32 


9-175362 


1436 


10-824638 


29 


32 


171389 


1402 


995165 


32 


176224 


1433 


823776 


28 


33 


172230 


1399 


995146 


32 


177084 


1431 


822916 


27 


34 


173070 


1396 


995127 


32 


177942 


1428 


822058 


26 


35 


173908 


1394 


995108 


32 


178799 


1425 


821201 


25 


36 


174744 


1391 


995089 


32 


179655 


1423 


820345 


24 


37 


175578 


1388 


995070 


32 


180508 


1420 


819492 


23 


38 


176411 


1386 


995051 


32 


181360 


1417 


818640 


22 


39 


177242 


1383 


995032 


32 


182211 


1415 


817789 


21 


40 


178072 


1380 


995013 


32 


183059 


1412 


816941 


20 


41 


9-178900 


1377 


9-994993 


32 


9-183907 


1409 


10-816093 


19 


42 


179726 


1374 


994974 


32 


184752 


J 407 


8 J 5248 


18 


43 


180551 


1372 


994955 


32 


185597 


1404 


814403 


17 


44 


181374 


1369 


994935 


32 


186439 


1402 


813561 


16 


45 


182196 


1366 


994916 


33 


187280 


1399 


812720 


15 


46 


183016 


1364 


994896 


33 


188120 


1396 


811880 


14 


47 


183834 


1361 


994877 


33 


188958 


1393 


811042 


13 


48 


184651 


1359 


994857 


33 


189794 


1391 


810206 


12 


49 


185466 


1356 


994838 


33 


190629 


1389 


809371 


11 


50 


186280 


1353 


994818 


33 


191462 


1386 


808538 


10 


51 


9-187092 


1351 


9-994798 


33 


9192294 


1384 


10-807706 


9 


52 


187903 


1348 


994779 


33 


193124 


1381 


806876 


8 


53 


188712 


1346 


994759 


33 


193953 


1379 


806047 


7 


54 


189519 


1343 


994739 


33 


194780 


1376 


805220 


6 


55 


190325 


1341 


994719 


33 


195606 


1374 


804394 


5 


56 


191130 


1338 


994700 


33 


196430 


1371 


803570 


4 


57 


191933 


1336 


994680 


33 


197253 


1369 


802747 


3 


58 


192734 


1333 


994660 


33 


198074 


1366 


801926 


2 


59 


193534 


1330 


994640 


33 


198894 


1364 


801106 


1 


60 


194332 


1328 


994620 


33 


199713 


1361 J 


800287 






| Cosine | | Sine | \ Cotang. 

81 Degrees. 



| Tan ff . | M 



LOGARITHMIC SIXES, COSINES, ETC. (9 Degrees.) 201 



i. 


1 Sine 


1 D. 


| Cosine 


1 D. 


Tan*. 


D. 


i Cotang: 







1 9-194332 


1328 


9-994620 


33 


9199713 


1361 


10-800287 


60 


1 


1 195129 


1326 


994600 


33 


200529 


1359 


799471 


59 


2 


195925 


1323 


994580 


33 


201345 


1356 


798655 


58 


3 


196719 


1321 


994560 


34 


202159 


J 354 


797841 


57 


4 


1 197511 


1318 


994540 


34 


202971 


1352 


797029 


56 


5 


198302 


1316 


994519 


34 


203782 


1349 


796218 


55 


6 


199091 


1313 


994499 


34 


204592 


1347 


795408 


54 


7 


199879 


1311 


994479 


34 


205400 


1345 


794600 


53 


8 


200666 


1308 


994459 


34 


206207 


1342 


793793 


52 


9 


201451 


1306 


994438 


34 


207012 


1340 


792987 


51 


10 


202234 


1304 


994418 


34 


207817 


1338 


792183 


50 


11 


0-203017 


1301 


9-994397 


34 


9-208619 


1335 


10-791381 


49 


2 


203797 


1299 


994377 


34 


209420 


1333 


790580 


48 


3 


204577 


1296 


994357 


34 


210220 


1331 


789780 


47 


4 


205354 


1294 


994336 


34 


211018 


1328 


788982 


46 


5 


206131 


1292 


994316 


34 


211815 


1326 


788185 


45 


6 


206906 


1289 


994295 


34 


212611 


1324 


787389 


44 


7 


207679 


1287 


994274 


35 


213405 


1321 


786595 


43 


8 


208452 


1285 


994254 


35 


214198 


1319 


785802 


42 


9 


209222 


1282 


994233 


35 


214989 


1317 


785011 


41 





209992 


1280 


994212 


35 


215780 


1315 


784220 


40 


M 


9210760 


1278 


9-994191 


35 


9-216568 


1312 


10-783432 


39 


►2 


211526 


1275 


994171 


35 


217356 


1310 


782644 


38 


3 


212291 


1273 


994150 


35 


218142 


1308 


781858 


37 


4 


213055 


1271 


994129 


35 


218926 


1305 


781074 


36 


5 


213818 


1268 


994108 


35 


219710 


1303 


780290 


35 


G 


214579 


1266 


994087 


35 


220492 


1301 


779508 


34 


7 


215338 


1264 


994066 


35 


221272 


1299 


778728 


33 


8 


216097 


1261 


994045 


35 


222052 


1297 


777948 


32 


9 


216854 


1259 


994024 


35 


222830 


1294 


777170 


31 





217609 


1257 


994003 


35 


223606 


1292 


776394 


30 


1 


9-218363 


1255 


9-993981 


35 


9-224382 


1290 


10-775618 


29 


2 


219116 


1253 


993960 


35 


225156 


1288 


774844 


28 


3 


219868 


1250 


993939 


35 


225929 


1286 


774071 


27 


4 


220618 


1248 


993918 


35 


226700 


1284 


773300 


26 


5 


221367 


1246 


993896 


36 


227471 


1281 


772529 


25 


G 


222115 


1244 


993875 


36 


228239 


1279 


771761 


24 


7 


222861 


1242 


993854 


36 


229007 


1277 


770993 


23 


8 


223606 


1239 


993832 


36 


229773 


1275 


770227 


22 


9 


224349 


1237 


993811 


36 


230539 


1273 


769461 


21 





225092 


1235 


993789 


36 


231302 


1271 


768698 


20 


1 


9-225833 


1233 


9-993768 


36 


9-232065 


1269 


10-767935 


19 


•2 


226573 


1231 


993746 


36 


232826 


1267 


767174 


18 


3 


227311 


1228 


993725 


36 


233586 


1265 


76G414 


17 


4 


228048 


1226 


993703 


36 


234345 


1262 


765655 


16 


5 


228784 


1224 


993681 


36 


235103 


1260 


764897 


15 


6 


229518 


1222 


993660 


36 


235859 


1258 


764141 


14 


7 


230252 


1220 


993638 


36 


236614 


1256 


763386 


13 


8 


230984 


1218 


993616 


36 


237368 


1254 


762632 


12 


9 


231714 


1216 


993594 


37 


238120 


1252 


761880 


11 





232444 


1214 


993572 


37 


238872 


1250 


761128 


10 


1 


9-233172 


1212 


9-993550 


37 


9-239622 


1248 


10-760378 


9 


2 


233899 


1209 


993528 


37 


240371 


1246 


759629 


8 


3 


234625 


1207 


993506 


37 


241118 


1244 


758882 


7 


4 


235349 


1205 


993484 


37 


241865 


1242 


758135 


6 


5 


236073 


1203 


993462 


37 


242610 


1240 


757390 


5 


G 


236795 


1201 


993440 


37 


243354 


1238 


756646 


4 


7 


237515 


1199 


993418 


37 


244097 


1236 


755903 


3 


3 


238235 


1197 


993396 


37 


244839 


1234 


7551G1 1 


2 


y 


238953 


1195 


993374 


37 


245579 


1232 


754421 1 


1 





239670 


1J93 


993351 


37 


246319 


1230 


753681 l 






| Cosine | 



| Sine 



I 



| Cotanff. | 



| Tang. I M. 



80 Degrees. 



208 (W Degrees.) 


LOGARITHMIC SINES, 


COSINES, ETO. 




M. 


| Sine 


J D. 


| Cosine 


1 D. 


f Tang. 


1 D. 


1 Cotang. 


; 





9-239670 


1193 


9-993351 


37 


9246319 


1230 


10-753681 


60 


1 


240386 


1191 


993329 


37 


247057 


1228 


752943 


59 


2 


241101 


1189 


993307 


37 


247794 


1226 


752206 


58 


3 


241814 


1187 


993285 


37 


248530 


1224 


751470 


57 


4 


242526 


1185 


993262 


37 


249264 


1222 


750736 


56 


5 


243237 


1183 


993240 


37 


249998 


1220 


750002 


55 


6 


243947 


1181 


993217 


38 


250730 


1218 


749270 


54 


7 


244656 


1179 


993195 


38 


251461 


1217 


748539 


53 


8 


245363 


1177 


993172 


38 


252191 


1215 


747809 


52 


9 


246069 


1175 


993149 


38 


252920 


1213 


747080 


51 


10 


246775 


1173 


993127 


38 


253648 


1211 


746352 


50 


11 


9-247478 


1171 


9-993104 


38 


9-254374 


1209 


10745626 


49 


12 


248181 


1169 


993081 


38 


255100 


1207 


744900 


48 


13 


248883 


1167 


993059 


38 


255824 


1205 


744176 


47 


14 


249583 


1165 


993036 


38 


256547 


1203 


743453 


46 


15 


250282 


1163 


993013 


38 


257269 


1201 


742731 


45 


16 


250980 


1161 


992990 


38 


257990 


1200 


742010 


44 


17 


251677 


1159 


992967 


38 


258710 


1198 


741290 


43 


18 


252373 


1158 


992944 


38 


259429 


1196 


740571 


42 


19 


253067 


1156 


992921 


38 


260146 


1194 


739854 


41 


20 


253761 


1154 


992898 


38 


260863 


1192 


739137 


40 


21 


9254453 


1152 


9-992875 


38 


9-261578 


1190 


10-738422 


39 


22 


255144 


1150 


992852 


38 


262292 


1189 


737708 


38 


23 


255834 


1148 


992829 


39 


263005 


1187 


736995 


37 


24 


256523 


1146 


992806 


39 


263717 


1185 


736283 


36 


25 


257211 


1144 


992783 


39 


264428 


1183 


735572 


35 


26 


257898 


1142 


992759 


39 


265138 


1181 


734862 


34 


27 


258583 


1141 


992736 


39 


265847 


1179 


734153 


33 


28 


259268 


1139 


992713 


39 


266555 


1178 


733445 


32 


29 


259951 


1137 


992690 


39 


267261 


1176 


732739 


31 


30 


260633 


1135 


992666 


39 


267967 


1174 


732033 


30 


31 


9-261314 


1133 


9-992643 


39 


9-268671 


1172 


10731329 


29 


32 


261994 


1131 


992619 


39 


269375 


1170 


73^025 


28 


33 


262673 


1130 


992596 


39 


270077 


1169 


729923 


27 


34 


263351 


1128 


992572 


39 


270779 


1167 


729221 


26 


35 


264027 


1126 


992549 


39 


271479 


1165 


728521 


25 


36 


264703 


1124 


992525 


39 


272178 


1164 


727822 


24 


37 


265377 


1122 


992501 


39 


272876 


1162 


727124 


23 


38 


266051 


1120 


992478 


40 


273573 


1160 


726427 


22 


39 


266723 


1119 


992454 


40 


274269 


1158 


725731 


21 


40 


267395 


1117 


992430 


40 


274964 


1157 


725036 


20 


41 


9-268065 


1115 


9-992406 


40 


9-275658 


1155 


10724342 


19 


42 


268734 


1113 


992382 


40 


276351 


1153 


723649 


IS 


43 


269402 


1111 


992359 


40 


277043 


1151 


722957 


17 


44 


270069 


1110 


992335 


40 


277734 


1150 


722266 


16 


45 


270735 


1108 


992311 


40 


278424 


1148 


721576 


15 


46 


271400 


1106 


992287 


40 


279113 


1147 


720887 


14 


47 


272064 


1105 


992263 


40 


279801 


1145 


720199 


13 


48 


272726 


1103 


992239 


40 


280488 


1143 


719512 


12 


49 


273388 


1101 


992214 


40 


281174 


1141 


718826 


11 


50 


274049 


1099 


992190 


40 


281858 


1140 


718142 


10 


51 


9-274708 


1098 


9-992166 


40 


9-282542 


1138 


10717458 


9 


52 


275367 


1096 


992142 


40 


283225 


1136 


716775 


8 


53 


276024 


1094 


992117 


41 


283907 


1135 


716093 


7 


54 


276681 


1092 


992093 


41 


284588 


1133 


715412 


6 


55 


277337 


1091 


992069 


41 


285268 


1131 


714732 


5 


56 


277991 


1089 


992044 


41 


285947 


1130 


714053 


4 


57 


278644 


1087 


992020 


41 


286624 


1128 


713376 


•> 


58 


279297 


1086 


991996 


41 


287301 


1126 


712:599 


2 


59 


279948 


1084 


991971 


41 


287977 


1125 


712023 


1 


60 1 


280599 


1082 


991947 


41 


288(152 


1123 l 


711348 





1 


Cosine | 


1 


Sine | 
79 


1 
Degree 


Cotang. | 

S. 


I 


Tang. ( 


M. 



LOGARITHMIC SINES, COSINES, ETC. (11 Degrees.) 209 



ft. 


) Sine 


1 D. 


| Cosine 


D. 


Tang. ' 


D. 


Cotang. 







9-28059!) 


1082 


9991947 


41 


9-288652 


1123 


10-711348 


60 


1 


281248 


1081 


991922 


41 


289326 


1122 


710674 


59 


2 


281897 


1079 


991897 


41 


289999 


1120 


710001 


58 


3 


282544 


1077 


991873 


41 


290671 


1118 


709329 


57 


4 


283190 


1076 


991848 


41 


291342 


1117 


708658 


56 


5 


283836 


1074 


991823 


41 


292013 


1115 


707987 


55 


6 


284480 


1072 


991799 


41 


292682 


1114 


707318 


54 


7 


285124 


1071 


991774 


42 


293350 


1112 


706650 


53 


8 


285766 


1069 


991749 


42 


294017 


1111 


705983 


52 


9 


286408 


1067 


991724 


42 


294684 


1109 


705316 


51 


10 


287048 


1066 


991699 


42 


295349 


1107 


704651 


50 


11 


9-287687 


1064 


9-991674 


42 


9-296013 


1106 


10-703987 


49 


12 


288326 


1063 


991649 


42 


296677 


1104 


703323 


48 


J 3 


288964 


1061 


991624 


42 


297339 


1103 


702661 


47 


\\ 


289600 


1059 


991599 


42 


298001 


1101 


701999 


46 


15 


290236 


1058 


991574 


42 


298662 


1100 


701338 


45 


10 


290870 


1056 


991549 


42 


299322 


1098 


700678 


44 


17 


291504 


1054 


991524 


42 


299980 


1096 


700020 


43 


18 


292137 


1053 


991498 


42 


300638 


1095 


699362 


42 


19 


292768 


1051 


991473 


42 


301295 


1093 


698705 


41 


20 


293399 


1050 


991448 


42 


301951 


1092 


698049 


40 


21 


9294029 


1048 


9-991422 


42 


9-302607 


1090 


10-697393 


39 


22 


294658 


1046 


991397 


42 


303261 


1089 


696739 


38 


23 


295286 


1045 


991372 


43 


303914 


1087 


696086 


37 


24 


295913 


1043 


991346 


43 


304567 


1086 


695433 


36 


25 


296539 


1042 


991321 


43 


305218 


1084 


694782 


35 


26 


297164 


1040 


991295 


43 


305869 


1083 


694131 


34 


27 


297788 


1039 


991270 


43 


306519 


1081 


693481 


33 


28 


298412 


1037 


991244 


43 


307168 


1080 


692832 


32 


29 


299034 


1036 


991218 


43 


307815 


1078 


692185 


31 


30 


299655 


1034 


991193 


43 


308463 


1077 


691537 


30 


31 


9-300276 


1032 


9991167 


43 


9-309109 


1075 


10-690891 


29 


32 


300895 


1031 


991141 


43 


309754 


1074 


690246 


28 


33 


301514 


1029 


991115 


43 


310398 


1073 


689602 


27 


34 


302132 


1028 


991090 


43 


311042 


1071 


688958 


26 


35 


302748 


1026 


991064 


43 


311685 


1070 


688315 


25 


36 


303364 


1025 


991038 


43 


312327 


1068 


687673 


24 


37 


303979 


• 1023 


991012 


43 


312967 


1067 


687033 


23 


38 


304593 


1022 


990986 


43 


313608 


1065 


686392 


22 


39 


305207 


1020 


990960 


43 


314247 


1064 


685753 


21 


40 


305819 


1019 


990934 


44 


314885 


1062 


6851 15 


20 


41 


9-306430 


1017 


9-990908 


44 


9 315523 


1061 


10-684477 


19 


42 


30704J 


1016 


990882 


44 


316159 


1060 


683841 


18 


43 


307650 


1014 


990855 


44 


316795 


1058 


683205 


17 


44 


308259 


1013 


990829 


44 


317430 


1057 


682570 


16 


45 


308867 


1011 


990803 


44 


318064 


1055 


681936 


15 


4(5 


309474 


1010 


990777 


44 


318697 


1054 


681303 


14 


47 


310080 


1008 


990750 


44 


319329 


1053 


680671 


13 


48 


310685 


1007 


990724 


44 


319961 


1051 


680039 


12 


49 


311289 


1005 


990697 


44 


320592 


1050 


679408 


11 


50 


311893 


1004 


990671 


44 


321222 


1048 


678778 


10 


51 


9312495 


1003 


9-990644 


44 


9-321851 


1047 


10-678149 


9 


52 


313097 


1001 


99061-8 


44 


322479 


1045 


C77521 


8 


53 


313698 


1000 


990591 


44 


323106 


1044 


676894 


7 


54 


314297 


998 


990565 


44 


323733 


1043 


676267 


C 


55 


314897 


997 


990538 


44 


3-24358 


1041 


675642 


5 


56 


315495 


996 


990511 


45 


324983 


1040 


675017 


4 


57 


316092 


994 


990485 


45 


325607 


1039 


674393 


3 


58 


316689 


993 


990458 


45 


326231 


1037 


673769 


2 


59 


317284 


991 


990431 


45 


326853 


1036 


673147 


1 


60 


317879 


990 


990404 


45 


327475 


1035 


672525 






( Cosine ( 



I 



| Cot an g. \ 



I Tang. | * 



78 Degrees. 



210 (12 Degrees.) LOGARITHMIC SINES, COSINES, ETC. 



M. | Sine 



D. | Cosine | D. | Tang. | D. | Cotang. 






9317879 


990 


9-990404 


45 


9-327474 


1035 


10672526 


60 


1 


318473 


988 


990378 


45 


328095 


1033 


671905 


59 


2 


319066 


987 


990351 


45 


328715 


1032 


671285 


58 


3 


319658 


986 


990324 


45 


329334 


1030 


670066 


57 


4 


320249 


984 


990297 


45 


329953 


1029 


670047 


56 


5 


320840 


983 


990270 


45 


330570 


1028 


669430 


55 


6 


321430 


982 


990243 


45 


331187 


1026 


668813 


54 


7 


322019 


980 


990215 


45 


331803 


1025 


6G8197 


53 


8 


322607 


979 


990188 


45 


332418 


1024 


667582 


52 


9 


323194 


977 


990161 


45 


333033 


1023 


666967 


51 


10 


323780 


976 


990134 


45 


333646 


1021 


666354 


50 


11 


9324366 


975 


9-990107 


46 


9-334259 


1020 


10-665741 


49 


12 


324950 


973 


990079 


46 


334871 


1019 


665129 


48 


13 


325534 


972 


990052 


46 


335482 


1017 


664518 


47 


14 


326117 


970 


990025 


46 


336093 


1016 


663907 


46 


15 


326700 


969 


989997 


46 


336702 


1015 


663298 


45 


1G 


327281 


968 


989970 


46 


337311 


1013 


662689 


44 


17 


327862 


966 


989942 


46 


337919 


1012 


662081 


43 


18 


328442 


965 


989915 


46 


338527 


1011 


661473 


42 


19 


329021 


964 


989887 


46 


339133 


1010 


660867 


41 


20 


329599 


962 


989860 


46 


339739 


1008 


660261 


40 


21 


9-330176 


961 


9-989832 


46 


9-340344 


1007 


10-659656 


39 


22 


330753 


960 


989804 


46 


340948 


1006 


659052 


38 


23 


331329 


958 


989777 


46 


341552 


1004 


658448 


37 


24 


331903 


957 


989749 


47 


342155 


1003 


657845 


36 


25 


332478 


956 


989721 


47 


342757 


1002 


657243 


35 


26 


333051 


954 


989693 


47 


343358 


1000 


656642 


34 


27 


333624 


953 


989665 


47 


343958 


999 


656042 


33 


28 


334195 


952 


989637 


47 


344558 


998 


655442 


32 


29 


334766 


950 


989609 


47 


345157 


997 


654843 


31 


30 


335337 


949 


989582 


47 


345755 


996 


654245 


30 


31 


9335906 


948 


9-989553 


47 


9-346353 


994 


10-653647 


29 


32 


336475 


946 


989525 


47 


346949 


993 


653051 


28 


33 


337043 


945 


989497 


47 


347545 


992 


652455 


27 


34 


337610 


944 


9894G9 


47 


348141 


991 


651859 


26 


35 


338176 


943 


989441 


47 


348735 


990 


651265 


25 


36 


338742 


941 


989413 


47 


349329 


988 


650671 


24 


37 


339306 


940 


989384 


47 


349922 


987 


' 650078 


23 


38 


339871 


939 


989356 


47 


350514 


986 


649486 


22 


39 


340434 


937 


989328 


47 


351106 


985 


648894 


21 


40 


340996 


936 


989300 


47 


351697 


983 


648303 


20 


41 


9-341558 


935 


9-989271 


47 


9-352287 


982 


10 647713 


19 


42 


342119 


934 


989243 


47 


352876 


981 


647124 


18 


43 


342679 


932 


989214 


47 


353465 


980 


646535 


17 


44 


343239 


931 


989186 


47 


354053 


979 


645947 


16 


45 


343797 


930 


989157 


47 


354640 


977 


645360 


15 


46 


344355 


929 


989128 


48 


355227 


976 


644773 


14 


47 


344912 


927 


989100 


48 


355813 


975 


644187 


13 


48 


345469 


926 


989071 


48 


356398 


974 


643602 


12 


49 


346024 


925 


989042 


48 


356982 


973 


643018 


U 


50 


346579 


924 


989014 


48 


357566 


971 


642434 


10 


51 


9-347134 


922 


9-988985 


48 


9358149 


970 


10-641851 


9 


52 


347687 


921 


988956 


48 


358731 


969 


641269 


8 


53 


348240 


920 


988927 


48 


359313 


968 


640687 


7 


54 


348792 


919 


988898 


48 


359893 


967 


640107 


6 


55 


349343 


917 


988869 


48 


360474 


966 


639526 


5 


56 


349893 


916 


988840 


48 


361053 


965 


638947 


4 


57 


350443 


915 


988811 


49 


361632 


963 


638368 


3 


58 


350992 


914 


988782 


49 


362210 


962 


637790 


2 


59 


351540 


913 


988753 


49 


362787 


961 


637213 


1 


60 


352088 


911 


988724 


49 


363364 


960 


636636 






I Cosine I 



Sine 1 | Cotang. | 

11 Degrees. 



| Tang. 



LOGARITHMIC SINES, COSINES, ETC. (13 Degrees.) 211 



M. | 



| D. | Cosine I 



t Tang. | D. | CoUng. 






9352088 


911 


9-988724 


49 


9363364 


960 


TO'636636 


60 


1 


352635 


910 


988695 


49 


363940 


959 


636060 


59 


2 


353181 


909 


988666 


49 


364515 


958 


635485 


58 


3 


353726 


908 


988636 


49 


365090 


957 


634910 


57 


4 


354271 


907 


988607 


49 


365664 


955 


634336 


56 


5 


354815 


905 


988578 


49 


366237 


954 


633763 


55 


8 


355358 


904 


988548 


49 


366810 


953 


633190 


54 


7 


355901 


903 


988519 


49 


367382 


952 


632618 


53 


8 


356443 


902 


988489 


49 


367953 


951 


632047 


52 


9 


356984 


901 


988460 


49 


368524 


950 


631476 


51 


10 


357524 


899 


988430 


49 


369094 


949 


630906 


50 


11 


9-358064 


898 


9-988401 


49 


9-369663 


948 


10*630337 


49 


12 


358603 


897 


988371 


49 


370232 


946 


629768 


48 


13 


359J41 


896 


988342 


49 


370799 


945 


629201 


47 


14 


359678 


895 


988312 


50 


371367 


944 


628633 


4o 


15 


360215 


893 


988282 


50 


371933 


943 


628067 


45 


16 


360752 


892 


988252 


50 


372499 


942 


627501 


44 


17 


361287 


891 


988223 


50 


373064 


941 


626936 


43 


18 


361822 


890 


988193 


50 


373629 


940 


626371 


42 


19 


362356 


889 


988163 


50 


374193 


939 


625807 


41 


20 


362889 


888 


988133 


50 


374756 


938 


625244 


40 


21 


9-363422 


887 


9-988103 


50 


9-375319 


937 


10-624681 


39 


22 


363954 


885 


988073 


50 


375881 


935 


624119 


38 


23 


364485 


884 


988043 


50 


376442 


934 


623558 


37 


24 


365016 


883 


988013 


50 


377003 


933 


622997 


36 


25 


365546 


882 


987983 


50 


377563 


932 


622437 


35 


26 


366075 


881 


987953 


50 


378122 


931 


621878 


34 


27 


366604 


880 


987922 


50 


378681 


930 


621319 


33 


28 


367131 


879 


987892 


50 


379239 


929 


620761 


32 


29 


367659 


877 


987862 


50 


379797 


928 


620203 


31 


30 


368185 


876 


987832 


51 


380354 


927 


619646 


30 


31 


9-368711 


875 


9987801 


51 


9-380910 


926 


10-619090 


29 


32 


369236 


874 


987771 


51 


381466 


925 


6 J 8534 


28 


33 


369761 


873 


987740 


51 


382020 


924 


617980 


27 


34 


370285 


872 


987710 


51 


382575 


923 


617425 


26 


35 


370808 


871 


987679 


51 


383129 


922 


616871 


25 


36 


371330 


870 


987649 


51 


383682 


921 


616318 


24 


37 


371852 


869 


987618 


"51 


384234 


920 


615766 


23 


38 


372373 


867 


987588 


51 


384786 


919 


615214 


22 


39 


372894 


866 


987557 


51 


385337 


918 


614663 


21 


40 


373414 


865 


987526 


51 


385888 


917 


614112 


20 


41 


9-373933 


864 


9-987496 


51 


9-386438 


915 


10-613562 


19 


42 


374452 


863 


987465 


51 


386987 


914 


613013 


18 


43 


374970 


862 


987434 


51 


387536 


913 


612464 


17 


44 


375487 


861 


987403 


52 


388084 


912 


611916 


16 


45 


376003 


860 


987372 


52 


388631 


91 J 


611369 


15 


46 


376519 


859 


987341 


52 


389178 


910 


610822 


14 


47 


377035 


858 


987310 


52 


389724 


909 


610276 


13 


48 


377549 


857 


987279 


52 


390270 


908 


609730 


12 


49 


378063 


856 


987248 


52 


390815 


907 


609185 


11 


50 


378577 


854 


987217 


52 


391360 


906 


608640 


10 


51 


9-379089 


853 


9-987186 


52 


9-391903 


905 


10-608097 


9 


52 


379601 


852 


987155 


52 


392447 


904 


607553 


8 


53 


380113 


851 


987124 


52 


392989 


903 


607011 


7 


54 


380624 


850 


987092 


52 


393531 


902 


606469 


6 


55 


381134 


849 


987061 


52 


394073 


901 


605927 


5 


56 


381643 


848 


987030 


52 


394614 


900 


605386 


4 


57 


382152 


847 


986*98 


52 


395154 


899 


604846 


3 


58 


382661 


846 


986967 


52 


395694 


898 


604306 


2 


59 


383168 


845 


986936 


52 


396233 


897 


603767 


1 


60 


383675 


844 


986904 


52 


396771 


896 


603229 






I Cceme | 



I Sine 



| Cotang. ) 



I Taqg. | M. 



76 Degrees. 



212 (14 Degrees.) LOGARITHMIC SINES, COSINES, ETC. 



M. 


Sine 


D. 


Cosine 


1 D. 


1 Tang. 


1 D. 


Cotang. 


1 





9383675 


844 


9-986904 


52 


9396771 


896 


10603229 


60 


I 


384 J 82 


843 


986873 


53 


397309 


896 


602G91 


59 


2 


384687 


842 


986841 


53 


397846 


895 


602154 


58 


3 


385192 


841 


986809 


53 


398383 


894 


601617 


57 


4 


385697 


840 


986778 


53 


398919 


893 


601081 


56 


5 


386201 


839 


986746 


53 


399455 


892 


600545 


55 


6 


386704 


838 


986714 


53 


399990 


891 


600010 


54 


7 


387207 


837 


986683 


53 


400524 


890 


599476 


53 


8 


387709 


836 


986651 


53 


401058 


889 


598942 


52 


9 


388210 


835 


986619 


53 


401591 


888 


598409 


51 


10 


388711 


834 


986587 


53 


402124 


887 


597876 


50 


11 


9-38921 1 


833 


9-986555 


53 


9-402656 


886 


10-597344 


49 


12 


389711 


832 


986523 


53 


403187 


885 


59C813 


48 


13 


390210 


831 


986491 


53 


403718 


884 


596282 


47 


H 


390708 


830 


986459 


53 


404249 


883 


595751 


46 


15 


391206 


828 


986427 


53 


404778 


882 


595222 


45 


16 


391703 


827 


986395 


53 


405308 


881 


594692 


44 


17 


392199 


826 


986363 


54 


405836 


880 


594164 


43 


18 


392695 


825 


986331 


54 


406364 


879 


593036 


42 


19 


393191 


824 


986299 


54 


406892 


878 


593108 


41 


20 


393685 


823 


986266 


54 


407419 


877 


592581 


40 


21 


9-394179 


822 


9-986234 


54 


9-407945 


876 


10-592055 


39 


22 


394673 


821 


986202 


54 


408471 


875 


591529 


38 


23 


395166 


820 


986109 


54 


408997 


874 


591003 


37 


24 


395658 


819 


986137 


54 


409521 


874 


590479 


36 


25 


396150 


818 


986104 


54 


410045 


873 


589955 


35 


26 


396641 


817 


986072 


54 


410569 


872 


589431 


34 


27 


397132 


817 


986039 


54 


411092 


871 


588908 


33 


28 


397621 


816 


986007 


54 


411615 


870 


588385 


32 


29 


398111 


815 


985974 


54 


412137 


869 


587863 


31 


30 


398600 


814 


985942 


54 


412658 


868 


587342 


30 


31 


9-399088 


813 


9-985909 


55 


9-413179 


867 


10-586821 


29 


32 


399575 


812 


985876 


55 


413699 


866 


586301 


28 


33 


400062 


811 


985843 


55 


414219 


865 


585781 


27 


34 


400549 


810 


985811 


55 


414738 


864 


585262 


26 


35 


401035 


809 


985778 


55 


415257 


864 


584743 


25 


36 


401520 


808 


985745 


55 


415775 


863 


584225 


24 


37 


402005 


807 


9857 J 2 


55 


416293 


862 


583707 


23 


38 


402489 


806 


985679 


55 


416810 


861 


583190 


22 


39 


402972 


805 


985646 


55 


417326 


860 


582674 


21 


40 


403455 


804 


985613 


55 


417842 


859 


582158 


20 


41 


9-403938 


803 


9-985580 


55 


9-418358 


858 


10-581642 


19 


42 


404420 


802 


985547 


55 


418873 


857 


581127 


18 


43 


404901 


801 


985514 


55 


419387 


856 


580613 


17 


44 


405382 


800 


985480 


55 


419901 


855 


580099 


16 


45 


405862 


799 


985447 


55 


420415 


855 


579585 


15 


46 


406341 


798 


985414 


56 


420927 


854 


579073 


14 


47 


406820 


797 


985380 


56 


421440 


853 


578560 


13 


48 


407299 


796 


985347 


56 


421952 


852 


578048 


12 


49 


407777 


795 


985314 


56 


422463 


851 


577537 


11 


50 


408254 


794 


985280 


56 


422974 


850 


577026 


10 


51 


9-408731 


794 


9-985247 


56 


9-423484 


849 


10-576516 


9 


52 


409207 


793 


985213 


56 


423993 


848 


576007 


8 


53 


409682 


792 


985180 


56 


424503 


848 


575497 


7 


54 


410157 


791 


985146 


56 


425011 


847 


574989 


6 


55 


410632 


790 


985113 


56 


425519 


846 


574481 


5 


56 


411106 


789 


985079 


56 


426027 


845 


573973 


4 


57 


41 1579 


788 


985045 


56 


426534 


844 


573466 


3 


58 


412052 


787 


985011 


56 


427041 


843 


572959 


2 


59 


412524 


786 


984978 


56 


427547 


843 


572453 


1 


60 


412996 


785 


984944 


56 


428052 


842 


571948 






| Cofin* | 



Sine | | Cotang. | 

75 Degrees. 



\ Tang. | M. 



LOGARITHMIC SINES, COSINES, ETC. (15 Degrees.) 213 



| Cosine | D. | Tan;*. 



D. | Cotang. | 






'9-412996 


785 


9-984944 


57 


9-428052 


842 


10-571948 


60 


1 


\ 413467 


784 


984910 


57 


428557 


841 


571443 


59 


2 


413938 


783 


984876 


57 


429062 


840 


570938 


58 


3 


414408 


783 


984842 


57 


429566 


839 


570434 


57 


4 


414878 


782 


984808 


57 


430070 


838 


569930 


56 


5 


415347 


781 


984774 


57 


430573 


838 


569427 


55 


6 


415815 


780 


984740 


57 


431075 


837 


568925 


54 


7 


416283 


779 


984706 


57 


431577 


836 


568423 


53 


8 


416751 


778 


984672 


57 


432079 


835 


567921 


52 


9 


417217 


777 


984637 


57 


432580 


834 


567420 


51 


10 


417684 


776 


984603 


57 


433080 


833 


566920 


50 


11 


9-418150 


775 


9-984569 


57 


9-433580 


832 


10-566420 


49 


12 


418615 


774 


984535 


57 


434080 


832 


565920 


48 


13 


419079 


773 


984500 


57 


434579 


831 


565421 


47 


14 


419544 


773 


984466 


57 


435078 


830 


564922 


46 


15 


420007 


772 


984432 


58 


435576 


829 


564424 


45 


16 


420470 


771 


984397 


58 


436073 


828 


563927 


44 


17 


420933 


770 


984363 


58 


436570 


828 


563430 


43 


18 


421395 


769 


984328 


58 


437067 


827 


562933 


42 


29 


421857 


768 


984294 


58 


437563 


826 


562437 


41 


20 


422318 


767 


984259 


58 


438059 


825 


561941 


40 


21 


9-422778 


767 


9-984224 


58 


9-438554 


824 


10-561446 


39 


22 


423238 


766 


984190 


58 


439048 


823 


560952 


38 


23 


423697 


765 


984155 


58 


439543 


823 


560457 


37 


24 


424156 


764 


984120 


58 


440036 


822 


559964 


36 


25 


424615 


763 


984085 


58 


440529 


821 


559471 


35 


26 


425073 


762 


984050 


58 


441022 


820 


558978 


34 


27 


425530 


761 


984015 


58 


441514 


819 


558486 


33 


28 


425987 


760 


983981 


58 


442006 


819 


557994 


32 


29 


426443 


760 


983946 


58 


442497 ' 


818 


557503 


31 


30 


426899 


759 


98391 1 


58 


442988 


817 


557012 


30 


31 


9-427354 


758 


9-983875 


58 


9-44341) 

443968 


816 


10-556521 


29 


32 


427809 


757 


983840 


59 


816 


556032 


28 


33 


428263 


756 


983805 


59 


444458 


815 


555542 


27 


34 


428717 


755 


983770 


59 


444947 


814 


555053 


26 


35 


429170 


754 


983735 


59 


445435 


813 


554565 


25 


36 


429623 


753 


983700 


59 


445923 


812 


554077 


24 


37 


430075 


752 


983664 


59 


446411 


812 


553589 


23 


38 


430527 


752 


983629 


59 


446898 


811 


553102 


22 


39 


430978 


751 


983594 


59 


447384 


810 


552616 


21 


40 


431429 


750 


983558 


59 


447870 


809 


552130 


20 


41 


9-431879 


749 


9-983523 


59 


9-448356 


809 


10-551644 


19 


42 


432329 


749 


983487 


59 


448841 


808 


551159 


18 


43 


432778 


748 


983452 


59 


449326 


807 


550674 


17 


44 


433226 


747 


983416 


59 


449810 


806 


550190 


16 


45 


433675 


746 


983381 


59 


450294 


806 


549706 


15 


46 


434122 


745 


983345 


59 


450777 


805 


549223 


14 


47 


434569 


744 


983309 


59 


451260 


804 


548740 


13 


48 


435016 


744 


983273 


60 


451743 


803 


548257 


12 


49 


. 435462 


743 


983238 


60 


452225 


802 


547775 


11 


50 


435908 


742 


983202 


60 


452706 


802 


547294 


10 


51 


9-436353 


741 


9-983166 


60 


9-453187 


801 


10-546813 


9 


52 


436798 


740 


983130 


60 


453668 


800 


546332 


8 


53 


437242 


740 


983094 


60 


454148 


799 


545852 


7 


54 


437686 


739 


983058 


60 


454628 


799 


545372 


6 


55 


438129 


738 


983022 


60 


455107 


798 


544893 


5 


56 


438572 


737 


982986 


60 


455586 


797 


544414 


4 


57 


439014 


736 


982950 


60 


456064 


796 


543936 


3 


58 


439456 


736 


982914 


60 


456542 


796 


543458 


2 


59 


439897 


735 


982878 


60 


457019 


795 


542981 


1 


60 


440338 


734 


982842 


60 


457496 


794 


542504 1 







Cosine \ 


/ 


Sine | 

7-4 


1 
Degree 


Cotang. | 
JS. 




Tang. | 


M. 



214 (16 Degrees.) 10GARITHMIC SINES, COSINES, ETC. 



M. | 



| Cosine | D. | Tang. ( D. 1 Cotang. | 






9-440338 


734 


9-982842 


60 


9-457496 


794 


10-542504 


66 


1 


440778 


733 


982805 


60 


457973 


793 


542027 


59 


2 


441218 


732 


982769 


61 


458449 


793 


541551 


58 


3 


441658 


731 


982733 


61 


458925 


792 


541075 


57 


4 


442096 


731 


982696 


61 


459400 


791 


540600 


56 


5 


442535 


730 


982660 


61 


459875 


790 


540125 


55 


6 


442973 


729 


982624 


61 


460349 


790 


539651 


51 


7 


443410 


728 


982587 


61 


460823 


789 


539177 


53 


8 


443847 


727 


982551 


61 


46J297 


788 


538703 


52 


9 


444284 


727 


982514 


61 


461770 


788 


538230 


51 


10 


44472,0 


726 


982477 


61 


462242 


787 


537758 


50 


11 


9445155 


725 


9-982441 


61 


9-462714 


786 


10-537286 


49 


12 


445590 


724 


982404 


61 


463186 


785 


536814 


48 


13 


446025 


723 


982367 


61 


463658 


785 


536342 


47 


14 


446459 


723 


982331 


61 


464129 


784 


535871 


46 


15 


446893 


722 


982294 


61 


464599 


783 


535401 


45 


16 


447326 


721 


982257 


61 


465069 


783 


534931 


44 


17 


447759 


720 


982220 


62 


465539 


782 


534461 


43 


18 


448191 


720 


982183 


62 


466008 


781 


533992 


12 


19 


448623 


719 


982146 


62 


466476 


780 


533524 


41 


20 


449054 


718 


982109 


62 


466945 


780 


533055 


40 


21 


9-449485 


717 


9-982072 


62 


9-467413 


779 


10-532587 


39 


22 


449915 


716 


982035 


62 


467880 


778 


532120 


38 


23 


450345 


716 


981998 


62 


468347 


778 


531653 


37 


24 


450775 


715 


981961 


62 


468814 


777 


531186 


36 


25 


451204 


714 


981924 


62 


469280 


776 


530720 


35 


26 


451632 


713 


981886 


62 


469746 


775 


530254 


34 


27 


452060 


713 


981849 


62 


470211 


775 


529789 


33 


28 


452488 


712 


981812 


62 


470676 


774 


529324 


32 


29 


452915 


711 


981774 


62 


471141 


773 


528859 


31 


30 


453342 


710 


981737 


62 


471605 


773 


528395 


30 


31 


9-453768 


710 


9-981699 


63 


9-472068 


772 


10-527932 


29 


32 


454194 


709 


981662 


63 


472532 


771 


527468 


28 


33 


454619 


708 


981625 


63 


472995 


771 


527005 


27 


34 


455044 


707 


981587 


63 


473457 


770 


526543 


26 


35 


455469 


707 


981549 


63 


473919 


769 


526081 


25 


36 


455893 


706 


981512 


63 


474381 


769 


525619 


24 


37 


456316 


705 


981474 


63 


474842 


768 


525158 


23 


38 


456739 


704 


981436 


63 


475303 


767 


524697 


22 


39 


457162 


704 


981399 


63 


475763 


767 


524237 


21 


40 


457584 


703 


981361 


63 


476223 


766 


523777 


20 


41 


9-458006 


702 


9-981323 


63 


9-476683 


765 


10-523317 


19 


42 


458427 


701 


981285 


63 


477142 


765 


522858 


18 


43 


458848 


701 


981247 


63 


477601 


764 


522399 


17 


44 


459268 


700 


981209 


63 


478059 


763 


521941 


16 


45 


459688 


699 


981171 


63 


478517 


763 


521483 


15 


46 


460108 


698 


981133 


64 


478975 


762 


521025 


14 


47 


460527 


698 


981095 


64 


479432 


761 


520568 


13 


48 


460946 


697 


981057 


64 


479889 


761 


520111 


12 


49 


461364 


696 


981019 


64 


480345 


760 


519655 


11 


50 


461782 


695 


980981 


64 


480801 


759 


519199 


10 


51 


9-462199 


695 


9-980942 


64 


9-481257 


759 


10-518743 


9 


52 


462616 


694 


980904 


64 


481712 


758 


5 J 8288 


8 


53 


463032 


693 


980866 


64 


482167 


757 


517833 


7 


54 


463448 


693 


980827 


64 


482621 


757 


517379 


6 


55 


463864 


692 


980789 


64 


483075 


756 


516925 


5 


56 


464279 


691 


980750 


64 


483529 


755 


516471 


4 


57 


464694 


690 


980712 


64 


483982 


755 


516018 


3 


58 


465108 


690 


980673 


64 


484435 


754 


515565 


2 


59 


465522 


689 


980635 


64 


484887 


753 


515113 


1 


80 


465935 


688 


980596 


64 


485339 


753 


514661 






\ Cosine 



Sine | I Cotang. | 

73 Degrees 



Tang. 



LOGARITHMIC SIXES, COSINES, ETC. (17 Degrees.) 215 



M. / 



1 Cosine \ D. 



Tang. / D. ( Cotang. | 






9-465935 


688 


9-980596 


64 


9-485339 


755 


10-514661 


60 


1 


466348 


688 


980558 


64 


485791 


752 


514209 


59 


2 


466761 


687 


980519 


65 


486242 


751 


513758 


58 


3 


467173 


686 


980480 


65 


486693 


751 


513307 


57 


4 


467585 


685 


980442 


65 


487143 


750 


512857 


56 


5 


467996 


685 


980403 


65 


487593 


749 


512407 


55 


6 


468407 


684 


980364 


65 


488043 


749 


511957 


54 


7 


468817 


683 


980325 


65 


488492 


748 


511508 


53 


S 


469227 


683 


980286 


65 


488941 


747 


511059 


52 


9 


469637 


682 


980247 


65 


489390 


747 


510610 


51 


IU 


470046 


681 


980208 


65 


489838 


746 


510162 


50 


11 


9-470455 


680 


9-980169 


65 


9-490286 


746 


10-509714 


49 


12 


470863 


680 


980130 


65 


490733 


745 


509267 


48 


13 


471271 


679 


980091 


65 


491180 


744 


508820 


47 


A 


471679 


678 


980052 


65 


491627 


744 


508373 


46 


15 


472086 


678 


980012 


65 


492C73 


743 


507927 


45 


1(3 


472492 


677 


979973 


65 


492519 


743 


507481 


44 


17 


472898 


676 


979934 


66 


492965 


742 


507035 


43 


18 


473304 


676 


979895 


66 


493410 


741 


506590 


42 


19 


473710 


675 


979855 


66 


493854 


740 


506146 


41 


20 


474115 


674 


979816 


66 


494299 


740 


505701 


40 


21 


9-474519 


674 


9-979776 


66 


9-494743 


740 


10-505257 


39 


22 


474923 


673 


979737 


66 


495186 


739 


504814 


38 


23 


475327 


672 


979697 


66 


495630 


738 


504370 


37 


24 


475730 


672 


979658 


66 


496073 


737 


503927 


36 


25 


476133 


671 


979618 


66 


496515 


737 


503485 


35 


26 


476536 


670 


979579 


66 


496957 


736 


503043 


34 


27 


476938 


669 


979539 


66 


497399 


736 


502601 


33 


25 


477340 


669 


979499 


66 


497841 


735 


502159 


32 


29 


477741 


668 


979459 


66 


498282 


734 


501718 


31 


30 


478142 


667 


979420 


66 


498722 


734 


501278 


30 


31 


9-478542 


667 


9-979380 


66 


9-499163 


733 


10-500837 


29 


32 


478942 


666 


979340 


66 


499603 


733 


500397 


28 


33 


479342 


665 


979300 


67 


500042 


732 


499958 


27 


34 


479741 


665 


979260 


67 


500481 


731 


499519 


26 


35 


480140 


664 


979220 


67 


500920 


731 


499C80 


25 


36 


480539 


663 


979180 


67 


601359 


730 


498641 


24 


37 


480937 


663 


979140 


67 


501797 


730 


498203 


23 


38 


481334 


662 


979100 


67 


502235 


729 


497705 


22 


39 


481731 


661 


979059 


67 


502672 


728 


497328 


21 


40 


482128 


661 


979019 


67 


503109 


728 


49G891 


20 


4i 


9-482525 


660 


9-978979 


67 


9-503546 


727 


10-496454 


19 


12 


482921 


659 


978939 


67 


503982 


727 


496018 


18 


43 


483316 


659 


978898 


67 


504418 


726 


495582 


17 


44 


483712 


658 


978858 


67 


504854 


725 


495146 


16 


45 


484107 


657 


978817 


67 


505289 


725 


494711 


15 


46 


484501 


657 


978777 


67 


505724 


724 


494276 


14 


47 


484895 


656 


978736 


67 


506159 


724 


493841 


13 


43 


485289 


655 


978696 


68 


506593 


723 


493407 


12 


49 


465682 


655 


978655 


68 


507027 


722 


4.92973 


11 


50 


486075 


654 


978615 


68 


507460 


722 


492540 


10 


51 


9-486467 


653 


9-978574 


68 


9-507893 


721 


10-492107 


9 


52 


486860 


653 


978533 


68 


5C8326 


721 


491674 


8 


53 


487251 


652 


9J8493 
978452 


68 


508759 


720 


491241 


7 


54 


487643 


651 


68 


509191 


719 


490809 


6 


55 


488034 


651 


978411 


68 


509622 


719 


490378 


5 


56 


488424 


650 


978370 


68 


510054 


718 


489946 


4 


57 


488814 


650 


978329 


68 


510485 


718 


489515 


3 


58 


489204 


649 


978288 


68 


510910 


717 


489084 


3 


5& 


489593 


648 


978247 


68 


511346 


716 


488654 


1 


60 


489982 


1 648 


978206 


68 


511776 


716 


1 488224 






; Cosine | 



I 



| Cotang. | 



{ Tariff. \ 



72 Degrees. 



216 (18 Degrees.) LOGARITHMIC SINES, COSINES, ETC. 



M. 


Sine 


D. | 


Cosine 


D. | 


Tang. | 


D. | 


Cotang. 







9-489982 


648 


9-978206 


68 


9-511776 


716 


10-488224 


60 


1 


490371 


648 


978165 


68 


512206 


716 


487794 


59 


2 


490759 


647 


978124 


68 


512635 


715 


487365 


58 


3 


491147 


646 


978083 


69 


513064 


714 


486936 


57 


4 


491535 


646 


978042 


69 


513493 


714 


486507 


56 


5 


491922 


645 


978001 


69 


513921 


713 


486079 


55 


6 


492308 


644 


977959 


69 


514349 


713 


485651 


54 


7 


492695 


644 


977918 


69 


514777 


712 


485223 


53 


8 


493081 


643 


977877 


69 


515204 


712 


484796 


52 


9 


493466 


642 


977835 


69 


515631 


711 


484369 


51 


10 


493851 


642 


977794 


69 


516057 


710 


483943 


50 


11 


9494236 


641 


9-977752 


69 


9-516484 


710 


10-483516 


49 


12 


494621 


641 


977711 


69 


516910 


709 


483090 


48 


13 


495005 


640 


977669 


69 


517335 


709 


482665 


47 


14 


495388 


639 


977628 


69 


517761 


708 


482239 


46 


15 


495772 


639 


977586 


69 


518185 


708 


481815 


45 


16 


496154 


638 


977544 


70 


518610 


707 


481390 


44 


17 


496537 


637 


977503 


70 


519034 


706 


480966 


43 


18 


496919 


637 


977461 


70 


519458 


706 


480542 


42 


19 


497301 


636 


977419 


70 


519882 


705 


480118 


41 


20 


497682 


636 


977377 


70 


520305 


705 


479695 


40 


21 


9-498064 


635 


9-977335 


70 


9*520728 


704 


10-479272 


39 


22 


498444 


634 


977293 


70 


521151 


703 


478849 


38 


23 


498825 


634 


977251 


70 


521573 


703 


478427 


37 


24 


499204 


633 


977209 


70 


521995 


703 


478005 


36 


25 


499584 


632 


977167 


70 


522417 


702 


477583 


35 


26 


499963 


632 


977125 


70 


522838 


702 


477162 


34 


27 


500342 


631 


977083 


70 


523259 


701 


476741 


33 


28 


500721 


631 


977041 


70 


523680 


701 


476320 


32 


29 


501099 


630 


976999 


70 


524100 


700 


475900 


31 


30 


501476 


629 


976957 


70 


524520 


699 


475480 


30 


31 


9-501854 


629 


9-970914 


70 


9-524939 


699 


10-475061 


29 


32 


502231 


628 


976872 


71 


525359 


698 


474641 


28 


33 


502607 


628 


976830 


71 


525778 


698 


474222 


27 


34 


502984 


627 


976787 


71 


526197 


697 


473803 


26 


35 


503360 


626 


976745 


71 


526615 


697 


473385 


25 


36 


503735 


626 


976702 


71 


527033 


696 


472967 


24 


37 


504110 


C25 


976660 


71 


527451 


696 


472549 


23 


38 


504485 


625 


970617 


71 


5278G8 


695 


472132 


22 


39 


504860 


624 


976574 


71 


528285 


695 


471715 


21 


40 


505234 


623 


976532 


71 


528702 


694 


471298 


20 


41 


9-505608 


623 


9-976489 


71 


9-529119 


693 


10-470881 


19 


42 


505981 


622 


970446 


71 


529535 


693 


470465 


18 


43 


506354 


622 


976404 


71 


529950 


693 


470050 


17 


44 


506727 


621 


976361 


71 


530366 


692 


469634 


16 


45 


507099 


620 


976318 


71 


530781 


691 


469219 


15 


46 


507471 


620 


976275 


71 


531196 


691 


468804 


14 


47 


507843 


619 


976232 


72 


531611 


690 


468389 


13 


48 


508214 


619 


976189 


72 


532025 


690 


467975 


12 


49 


508585 


618 


976146 


72 


532439 


689 


467561 


11 


50 


508956 


618 


"476103 


72 


532853 


689 


467147 


10 


51 


9509326 


617 


9 976060 


72 


9-533266 


688 


10-466734 


9 


52 


509696 


616 


976017 


72 


533679 


688 


466321 


8 


53 


510065 


616 


975974 


72 


534092 


687 


465908 


7 


54 


510434 


615 


975930 


72 


534^)4 


687 


465496 


6 


55 


510803 


615 


975887 


72 


534916 


686 


465084 


5 


56 


511172 


614 


975844 


72 


535328 


686 


464672 


4 


57 


511540 


613 


975800 


72 


535739 


685 


464261 


3 


58 


511907 


613 


975757 


72 


536150 


685 


463850 


o 


59 


512275 


612 


975714 


72 


536561 


684 


463439 


1 


60 


512642 


| 612 


| 975670 


72 


536972 


| 684 


463028 


1 o 



I Cosine | 



Sine 



I I L Cotang. | 

71 Degrees. 



I Tang. 



LOGARITHMIC SINES, COSINES, ETC. (19 Degrees.) 217 



M. | 


Sine f 


D. | 


Cosine | 


D. | 


Tan&. 


D. i 


Cotang. | 







9 5J2642 


612 


9975670 


73 


9-536972 


684 


10-463028 


60 


1 


513009 


611 


975627 


73 


537382 


683 


462618 


59 


2 


513375 


611 


975583 


73 


537792 


683 


462208 


58 


3 


513741 


610 


975539 


73 


538202 


682 


461798 


57 


4 


514107 


609 


975496 


73 


538611 


682 


461389 


56 


5 


514472 


609 


975452 


73 


539020 


681 


460980 


55 


6 


514837 


608 


975408 


73 


539429 


681 


460571 


54 


7 


515202 


608 


975365 


73 


539837 


680 


460163 


53 


8 


515566 


607 


975321 


73 


540245 


680 


459755 


52 


9 


515930 


607 


975277 


73 


540653 


679 


459347 


51 


10 


516294 


606 


975233 


73 


541061 


679 


458939 


50 


11 


9-516657 


605 


9-975189 _ 


73 


9-541468 


678 


10-458532 


49 


12 


517020 


605 


975145 


73 


541875 


678 


458125 


48 


13 


517382 


604 


975101 


73 


542281 


677 


457719 


47 


14 


517745 


604 


975057 


73 


542688 


677 


457312 


46 


15 


518107 


603 


975013 


73 


543094 


676 


456906 


45 


16 


518468 


603 


974969 


74 


543499 


676 


456501 


44 


17 


518829 


602 


974925 


74 


543905 


675 


456095 


43 


18 


519190 


601 


974880 


74 


544310 


675 


455690 


42 


19 


519551 


601 


974836 


74 


544715 


674 


455285 


41 


20 


519911 


600 


974792 


74 


545119 


674 


454881 


40 


21 


9520271 


600 


9-974748 


74 


9-545524 


673 


10-454476 


39 


22 


520631 


599 


974703 


74 


545928 


673 


454072 


38 


23 


520990 


599 


974659 


74 


546331 


672 


453669 


37 


24 


521349 


598 


974614 


74 


546735 


672 


453265 


36 


25 


521707 


593 


974570 


74 


547138 


671 


452862 


35 


26 


522066 


597 


974525 


74 


547540 


671 


452460 


34 


27 


522424 


596 


974481 


74 


547943 


670 


452057 


33 


28 


522781 


596 


974436 


74 


548345 


670 


451655 


32 


2& 


523138 


595 


974391 


74 


548747 


669 


451253 


31 


30 


523495 


595 


974347 


75 


549149 


669 


450851 


30 


31 


9-523852 


594 


9-974302 


75 


9-549550 


668 


10-450450 


29 


32 


524208 


594 


974257 


75 


549951 


668 


450049 


28 


33 


524564 


593 


974212 


75 


550352 


667 


449648 


27 


34 


524920 


593 


974167 


75 


550752 


667 


449248 


26 


35 


525275 


592 


974122 


75 


551152 


666 


448848 


25 


36 


525630 


591 


974077 


75 


551552 


666 


448448 


24 


37 


525984 


591 


974032 


75 


551952 


665 


448048 


23 


38 


526339 


590 


973987 


75 


552351 


665 


447649 


22 


39 


526693 


590 


973942 


75 


552750 


665 


447250 


21 


40 


527046 


589 


973897 


75 


553149 


664 


446851 


20 


41 


9-527400 


589 


9-973852 


75 


9-553548 


664 


10-446452 


19 


42 


627753 


588 


973807 


75 


553946 


663 


446054 


18 


43 


528105 


588 


973761 


75 


554344 


663 


445656 


17 


44 


528458 


587 


973716 


76 


554741 


662 


445259 


16 


45 


528810 


587 


973671 


76 


555139 


662 


444861 


15 


46 


529161 


586 


973625 


76 


555536 


661 


444464 


14 


47 


529513 


586 


973580 


76 


555933 


661 


444067 


13 


48 


529864 


585 


973535 


76 


556329 


660 


443671 


12 


49 


530215 


585 


973489 


76 


556725 


660 


443275 


11 


50 


530565 


584 


973444 


76 


557121 


659 


442879 


10 


51 


9 530915 


584 


9-973398 


76 


9-557517 


659 


10-442483 


3 


52 


531265 


583 


973352 


76 


557913 


659 


442087 


8 


53 


531614 


582 


973307 


76 


558308 


658 


441692 


7 


54 


531963 


582 . 


973261 


76 


558702 


658 


441298 


6 


55 


532312 


581 


973215 


76 


559097 


657 


440903 


5 


56 


532661 


581 


973169 


76 


559491 


657 


440509 


4 


57 


533009 


580 


973124 


76 


559885 


656 


440115 


3 


58 


533357 


580 


973078 


76 


560279 


656 


439721 


2 


59 


533704 


579 


973032 


77 


560673 


655 


439327 


1 


60 


534052 


578 


1 972986 


1 77 


561066 


655 


438934 







1 Cosine 




i iin« 

7( 


1 

) Degre 


| Cotang. 

es. 


1 


1 Tang. 


H 



218 (20 Degrees.) LOGARITHMIC SINES, COSINES, ETC. 



M. | 



| Cosine | 



| Tang. | D. | Cotang. | 






9534052 


578 


9-972986 


77 


9561066 


655 


10-438034 


60 


1 


534399 


577 


972940 


77 


561459 


654 


438541 


59 


2 


534745 


577 


972894 


77 


561851 


654 


438149 


58 


3 


535092 


577 


972848 


77 


562244 


653 


437756 


57 


4 


535438 


576 


972802 


77 


562636 


653 


437364 


56 


5 


535783 


576 


972755 


77 


563028 


653 


436972 


55 


6 


536129 


575 


972709 


77 


563419 


652 


436581 


54 


7 


536474 


574 


972663 


77 


563811 


652 


436189 


53 


8 


536818 


574 


972617 


77 


564202 


651 


435798 


52 


9 


537163 


573 


972570 


77 


564592 


651 


435408 


51 


JO 


537507 


573 


972524 


77 


564983 


650 


435017 


50 


11 


9537851 


572 


9-972478 


77 


9565373 


650 


10-434627 


49 


12 


538194 


572 


972431 


18 


565763 


649 


434237 


48 


13 


538538 


571 


972385 


78 


566153 


649 


433847 


47 


14 


538880 


571 


972338 


78 


566542 


649 


433458 


46 


15 


539223 


570 


972291 


78 


566932 


648 


433068 


45 


16 


539565 


570 


972245 


78 


567320 


648 


432680 


44 


17 


539907 


569 


972198 


78 


567709 


647 


432291 


43 


18 


540249 


569 


972151 


78 


568098 


647 


431902 


42 


19 


540590 


568 


972105 


78 


568486 


646 


431514 


41 


20 


540931 


568 


972058 


78 


568873 


646 


431127 


4( 


21 


9-541272 


567 


9-972011 


78 


9-569261 


645 


10-430739 


3& 


22 


541613 


567 


971964 


78 


569648 


645 


430352 


38 


23 


541953 


566 


971917 


78 


570035 


645 


429965 


37 


24 


542293 


566 


971870 


78 


570422 


644 


429578 


36 


25 


542632 


565 


971823 


78 


570809 


644 


429191 


35 


26 


542971 


565 


971776 


78 


571195 


643 


428805 


34 


27 


543310 


564 


971729 


79 


571581 


643 


428419 


33 


28 


543649 


564 


971682 


79 


571967 


642 


428033 


32 


29 


543987 


563 


971635 


79 


572352 


642 


427648 


31 


30 


544325 


563 


971588 


79 


572738 


642 


427262 


30 


31 


9-544663 


562 


9-971540 


79 


9-573123 


641 


10-426877 


29 


32 


545000 


562 


971493 


79 


573507 


641 


426493 


28 


33 


545338 


561 


971446 


79 


573892 


640 


426108 


27 


34 


545674 


561 


971398 


79 


574276 


640 


425724 


26 


35 


546011 


560 


971351 


79 


574660 


639 


425340 


25 


36 


546347 


560 


971303 


79 


575044 


639 


424956 


24 


37 


546683 


559 


971256 


79 


575427 


639 


424573 


23 


38 


547019 


559 


971208 


79 


575810 


638 


424190 


22 


39 


547354 


558 


971161 


79 


576193 


638 


423807 


21 


40 


547689 


558 


971113 


79 


576576 


637 


423424 


20 


41 


9548024 


557 


9-971066 


80 


9-576958 


637 


10-423041 


19 


42 


548359 


557 


971018 


80 


577341 


638 


422659 


18 


43 


548693 


556 


970970 


8C 


577723 


636 


422277 


17 


44 


549027 


556 


970922 


80 


578104 


636 


421896 


16 


45 


549360 


555 


970874 


80 


578486 


635 


421514 


15 


46 


549693 


555 


970827 


80 


578867 


635 


421133 


14 


47 


550026 


554 


970779 


80 


579248 


634 


420752 


13 


48 


550359 


554 


970731 


80 


579629 


634 


420371 


12 


49 


550692 


553 


970683 


80 


580009 


634 


419991 


11 


50 


551024 


553 


970635 


80 


580389 


633 


419611 


10 


51 


<)-551356 


552 


9-970586 


80 


9-580769 


633 


10-419231 


9 


52 


551687 


552 


970538 


80 


581149 


632 


418851 


8 


53 


552018 


552 


970490 


80 


581528 


632 


418472 


7 


54 


552349 


551 


970442 


80 


581907 


632 


418093 


6 


55 


552680 


551 


970394 


80 


582286 


631 


417714 


5 


56 


553010 


550 


970345 


81 


582665 


631 


417335 


4 


57 


553341 


550 


970297 


81 


583043 


630 


41 ("957 


3 


58 


553670 


549 


970249 


81 


583422 


630 


41C578 


2 


59 


554000 


549 


970200 


81 


583800 


629 


4K-200 


1 


60 


554329 


548 


970152 


81 


584177 


629 


415823 






| Cosine \ 



I 



| Cotang. ] 



I Tang. | iL 



69 Degrees, 



LOGARITHMIC SINES, COSINES, ETC. (21 Degrees.) 219 



H. 


Sine 


D. 


Cosine 


D. 


Tang. 


D. 


Cotang. 







9554329 


543 


9-970152 


81 


9-584177 


629 


10-415823 


60 


1 


554658 


548 


970103 


81 


584555 


629 


415445 


59 


2 


554987 


547 


9~0055 


81 


584932 


628 


415068 


58 


3 


555315 


547 


970006 


81 


585309 


628 


414691 


57 


4 


555643 


546 


969957 


81 


585686 


627 


414314 


56 


5 


555971 


546 


969909 


81 


586062 


627 


413938 


55 


6 


556299 


545 


909860 


81 


586439 


627 


413501 


54 


7 


556626 


545 


969811 


81 


586815 


626 


413185 


53 


8 


556953 


544 


969762 


81 


567190 


626 


412810 


52 


9 


557280 


544 


969714 


81 


587566 


625 


412434 


51 


10 


557606 


543 


969065 


81 


587941 


625 


412059 


50 


11 


9 557932 


543 


9-969616 


82 


9-588316 


625 


10-411684 


49 


12 


558258 


543 


969567 


82 


588691 


624 


411309 


48 


13 


558583 


542 


969518 


82 


589066 


624 


41 0934 


47 


11 


558909 


542 


969469 


82 


589440 


623 


41C560 


46 


15 


559234 


541 


969420 


82 


589814 


623 


410186 


45 


16 


559558 


541 


969370 


82 


590188 


623 


4C9812 


44 


17 


559883 


540 


969321 


82 


590562 


622 


409438 


43 


18 


560207 


540 


969272 


82 


590935 


622 


409065 


42 


19 


560531 


539 


969223 


82 


591308 


622 


4C8692 


41 


20 


560855 


539 


969173 


82 


591681 


621 


408319 


4a 


21 


9-561178 


538 


9-969124 


82 


9-592054 


621 


10-407946 


39 


22 


561501 


538 


969075 


82 


592426 


620 


407574 


38^ 


23 


561824 


537 


969025 


82 


592798 


620 


407202 


37 


24 


562146 


537 


968976 


82 


593170 


619 


406829 


361 


25 


562468 


536 


968926 


83 


593542 


619 


406458 


3£ 


26 


562790 


536 


968877 


83 


593914 


618 


406086 


34 


27 


563112 


536 


968827 


83 


594285 


618 


405715 


33 


2? 


563433 


535 


968777 


83 


594656 


618 


405344 


32 


29 


563755 


535 


968728 


83 


595027 


617 


404973 


31 


30 


564075 


534 


968678 


83 


595398 


617 


4046C2 


3C 


31 


9-564396 


534 


9-968628 


83 


9-595768 


617 


10-404232 


29- 


32 


5H47J6 


533 


968578 


83 


596138 


616 


403862 


28 


33 


565036 


533 


968528 


83 


596508 


616 


403492 


27 


34 


565356 


532 


968479 


83 


596878 


616 


403122 


26 


35 


565676 


532 


968429 


83 


597247 


615 


402753 


25 


36 


565995 


531 


968379 


83 


597616 


615 


402384 


24 


37 


566314 


531 


968329 


S3 


597985 


615 


402015 


23 


38 


566632 


531 


968278 


83 


598354 


614 


401646 


22 


39 


566951 


530 


968228 


84 


598722 


614 


401278 


21 


40 


567269 


530 


968178 


84 


599091 


613 


400909 


20 


41 


9-567587 


529 


9-968128 


84 


9-599459 


613 


10-400541 


19 


42 


567904 


529 


968078 


84 


599827 


613 


400173 


18 


43 


568222 


528 


968027 


84 


600194 


612 


399806 


17 


44 


568539 


528 


967977 


84 


600562 


612 


399438 


16 


45 


568856 


528 


967927 


84 


600929 


611 


399071 


15 


46 


569172 


527 


967876 


84 


601296 


611 


398704 


14 


47 


569488 


527 


967826 


84 


601662 


611 


398338 


13 


48 


569804 


526 


967775 


84 


602029 


610 


397971 


12 


49 


570120 


526 


967725 


84 


602395 


610 


397605 


11 


50 


570435 


525 


967674 


84 


602761 


610 


397239 


10 


51 


9-570751 


525 


9-967624 


84 


9-603127 


609 


10-396873 


9 


52 


571066 


524 


967573 


84 


603493 


609 


396507 


8 


53 


571380 


5^4 


967522 


85 


603858 


609 


396142 


7 


54 


571695 


523 


967471 


85 


604223 


608 


395777 


6 


55 


572009 


523 


967421 


85 


604588 


608 


395412 


5 


56 


572323 


523 


967370 


85 


604953 


607 


395047 


4 


57 


572636 


522 


967319 


85 


605317 


607 


394683 


3 


58 


572950 


522 


967268 


85 


605682 


607 


394318 


2 


59 


573263 


521 


967217 


85 


606046 


606 


393954 


1 


60 


573575 


521 


967166 


85 


606410 


606 


393590 






t «3p«ue j 



I Twg. | M. 



68 Degrees. 



(22 Degrees.) LOGARITHMIC SINES, COSINES, ETC. 



M. 


Sine 


1 D. 


Cosine 


1 D- 


\ Tang. 


1 D. 


| Cotang. 







9573575 


521 


9-967166 


85 


9606410 


606 


10393590 


69 


1 


573888 


520 


967115 


85 


606773 


606 


393227 


59 


2 


574200 


520 


967064 


85 


607137 


605 


392863 


58 


3 


574512 


519 


967013 


85 


607500 


605 


392500 


57 


4 


574824 


519 


966961 


85 


607863 


604 


392137 


56 


5 


575136 


519 


966910 


85 


608225 


604 


391775 


55 


6 


575447 


518 


966859 


85 


608588 


604 


391412 


54 


7 


575758 


518 


966808 


85 


608950 


603 


391050 


53 


8 


576069 


517 


966756 


86 


609312 


603 


390688 


52 


9 


576379 


517 


966705 


86 


609674 


603 


390326 


51 


10 


576689 


516 


966653 


86 


610036 


602 


389964 


50 


11 


9-576999 


516 


9-966602 


86 


9-610397 


602 


10-389603 


49 


12 


577309 


516 


966550 


86 


610759 


602 


389241 


48 


13 


577618 


515 


966499 


86 


611120 


601 


388880 


47 


14 


577927 


515 


966447 


86 


611480 


601 


388520 


46 


15 


578236 


514 


966395 


86 


611841 


601 


388159 


45 


16 


578545 


514 


966344 


86 


612201 


600 


387799 


44 


17 


578853 


513 


966292 


86 


612561 


600 


387439 


43 


18 


579162 


513 


966240 


86 


612921 


600 


387079 


42 


19 


579470 


513 


966188 


86 


613281 


599 


386719 


41 


20 


579777 


512 


966136 


86 


613641 


599 


386359 


40 


21 


9580085 


512 


9 966085 


87 


9614000 


598 


10-386000 


39 


22 


580392 


511 


966033 


87 


614359 


598 


385641 


38 


23 


580699 


511 


965981 


87 


614718 


598 


385282 


37 


24 


581005 


511 


965928 


87 


615077 


597 


384923 


36 


25 


581312 


510 


965876 


87 


615435 


597 


384565 


35 


2G 


581618 


510 


965824 


87 


615793 


597 


384207 


34 


27 


• 581924 


509 


965772 


87 


616151 


596 


383849 


33 


28 


582229 


509 


965720 


87 


616509 


596 


383491 


32 


29 


582535 


509 


965668 


87 


616867 


596 


383133 


31 


30 


582840 


508 


965615 


87 


617224 


595 


382776 


30 


31 


9-583145 


508 


9-965563 


87 


9617582 


595 


10-382418 


29 


32 


583449 


507 


965511 


87 


617939 


595 


382061 


28 


33 


583754 


507 


965458 


87 


618295 


594 


381705 


27 


34 


584058 


506 


965406 


87 


618652 


594 


381348 


26 


35 


584361 


506 


965353 


88 


619008 


594 


380992 


25 


36 


584665 


506 


965301 


88 


619364 


593 


380636 


24 


37 


584968 


505 


965248 


88 


619721 


593 


380279 


23 


38 


585272 


505 


965195 


88 


620076 


593 


379924 


22 


39 


585574 


504 


965143 


88 


620432 


592 


379568 


21 


40 


585877 


504 


965090 


88 


620787 


592 


379213 


20 


41 


9-586179 


503 


9-965037 


88 


9-621142 


592 


10-378858 


19 


42 


586482 


503 


964984 


88 


621497 


591 


378503 


18 


43 


586783 


503 


964931 


88 


621852 


591 


378148 


17 


44 


587085 


502 


964879 


88 


622207 


590 


377793 


16 


45 


587386 


502 


964826 


88 


622561 


590 


377439 


15 


46 


587688 


501 


964773 


88 


622915 


590 


377085 


14 


47 


587989 


501 


964719 


88 


623269 


589 


376731 


13 


48 


588289 


501 


964666 


89 


623623 


589 


376377 


12 


49 


588590 


500 


964613 


89 


623976 


589 


376024 


11 


50 


588890 


500 


964560 


89 


624330 


588 


375670 


10 


51 


9-589190 


499 


9-964507 


89 


9-624683 


588 


10375317 


9 


52 


589489 


499 


964454 


89 


625036 


588 


374964 


8 


53 


589789 


499 


964400 


89 


625388 


587 


374612 


7 


54 


590088 


498 


964347 


89 


625741 


587 


374259 


6 


55 


590387 


498 


964294 


89 


626093 


587 


373907 


5 


56 


590686 


497 


964240 


89 


626445 


586 


373555 


4 


57 


590984 


497 


964187 


89 


626797 


586 


373203 


3 


58 


591282 


497 


964133 


89 


627149 


586 


372851 


2 


59 


591580 


496 


964080 


89 


627501 


585 


372499 


1 


60 


591878 


496 


964026 


89 


627852 


585 


372148 






I Cosine | 



Sine | | Cotang. J 

67 Degrees. 



1 Tangr. I 



LOGARITHMIC SIXES, COSINES, ETC. (23 Degrees.) 221 



M. 


Sine | 


D. I 


Cosine 


D. 


Tang. | 


D. | 


Cotang. j 







9591878 


496 


9-964026 


89 


9-627852 1 


585 


10-372148 


60 


1 


592176 


495 


963972 


89 


628203 


585 


371797 


59 


2 


592473 


495 


963919 


89 


628554 


585 


371446 


58 


3 


592770 


495 


963865 


90 


628905 


584 


371095 


57 


4 


593067 


494 


963811 


90 


629255 


584 


370745 


56 


5 


593363 


494 


963757 


90 


629606 


583 


370394 


55 


6 


593659 


493 


963704 


90 


629956 


583 


370044 


54 


7 


593955 


493 


963650 


90 


630306 


583 


369694 


53 


8 


594251 


493 


963596 


90 


630656 


583 


369344 


52 


9 


594547 


492 


963542 


90 


631005 


582 


368995 


51 


10 


594842 


492 


963488 


90 


631355 


582 


368645 


50 


11 


9-595137 


491 


9-963434 


90 


9-631704 


582 


10-368296 


49 


12 


595432 


491 


963379 


90 


632053 


581 


367947 


48 


13 


595727 


491 


963325 


90 


632401 


581 


367599 


47 


14 


596021 


490 


963271 


90 


632750 


581 


367250 


46 


15 


596315 


490 


963217 


90 


633098 


580 


366902 


45 


J? 


596609 


489 


963163 


90 


633447 


580 


366553 


44 


596903 


489 


963108 


91 


633795 


580 


366205 


43 


18 


597196 


489 


963054 


91 


634143 


579 


365857 


42 


19 


597490 


488 


962999 


91 


634490 


579 


365510 


41 


20 


597783 


488 


962945 


91 


634838 


579 


365162 


40 


21 


9-598075 


487 


9-962890 


91 


9-635185 


578 


10-364815 


39 


22 


598368 


487 


962836 


91 


635532 


578 


364468 


38 


23 


598660 


487 


962781 


91 


635879 


578 


364121 


37 


24 


598952 


486 


962727 


91 


636226 


577 


363774 


36 


25 


599244 


486 


962672 


91 


636572 


577 


363428 


35 


26 


599536 


485 


962617 


91 


636919 


577 


363081 


34 


27 


599827 


485 


962562 


91 


637265 


577 


362735 


33 


28 


600118 


485 


962508 


91 


637611 


576 


362389 


32 


29 


600409 


484 


962453 


91 


637956 


576 


362044 


31 


30 


600700 


484 


962398 


9t 


638302 


576 


361698 


30 


31 


9-600990 


484 


9-962343 


92 


9-638647 


575 


10361353 


29 


32 


601280 


483 


962288 


92 


638992 


575 


361008 


28 


33 


601570 


483 


962233 


92 


639337 


575 


360663 


27 


34 


601860 


482 


962178 


92 


639682 


574 


360318 


26 


35 


602150 


482 


962123 


92 


640027 


574 


359973 


25 


36 


602439 


482 


962067 


^2 


640371 


574 


359629 


24 


37 


602728 


481 


962012 


92 


640716 


573 


359284 


23 


38 


603017 


481 


961957 


92 


641060 


573 


358940 


22 


39 


603305 


481 


961902 


92 


641404 


573 


358596 


21 


10 


603594 


480 


961846 


92 


641747 


572 


358253 


20 


41 


9-603882 


480 


9-961791 


92 


9-642091 


572 


10357909 


19 


42 


604170 


479 


961735 


92 


642434 


572 


357566 


18 


43 


604457 


479 


961680 


92 


642777 


572 


357223 


17 


44 


604745 


479 


961624 


93 


643120 


571 


356880 


16 


45 


605032 


478 


961569 


93 


643463 


571 


356537 


15 


46 


605319 


478 


961513 


93 


643806 


571 


356194 


14 


47 


605606 


478 


961458 


93 


644148 


570 


355852 


13 


48 


605892 


477 


961402 


93 


644490 


570 


355510 


12 


49 


606179 


4.n 


961346 


93 


644832 


570 


355168 


11 


50 


606465 


476 


961290 


93 


645174 


569 


354826 


10 


51 


9-606751 


476 


9-961235 


93 


9-645516 


569 


10-354484 


9 


52 


607036 


476 


961179 


93 


645857 


569 


354143 


8 


53 


607322 


475 


961123 


93 


646199 


569 


353801 


7 


54 


607607 


475 


961067 


93 


646540 


568 


353460 


6 


55 


607892 


474 


961011 


93 


646881 


568 


353119 


5 


56 


608177 


474 


960955 


93 


647222 


568 


352778 


4 


57 


608461 


474 


960899 


93 


647562 


567 


352438 


3 


58 


608745 


473 


960843 


94 


647903 


567 


352097 


o 


59 


609029 


473 


960786 


94 


648243 


567 


351757 


1 


60 


609313 


473 


960730 


94 


1 648583 


566 


351417 


(J 



Cosine 1 



| Sine | 



| Cotang. | 



| Tang. | M. 



66 Degrees. 



222 (24 Degrees.) LOGARITHMIC SINES, COS WES, ETC. 



M. | 


Sme 


D. \ 


Cosine ] 


D. fr 


Tang. | 


D. 


Cotang. 







9609313 


473 


9-960730 


94 


9-648583 


566 


10351417 


60 


1 


609597 


472 


960674 


94 


648923 


566 


351077 


59 


2 


609880 


472 


960618 


94 


649263 


566 


350737 


58 


3 


610164 


472 


960561 


94 


649602 


566 


350398 


57 


4 


610447 


471 


960505 


94 


649942 


565 


350058 


56 


5 


610729 


471 


960448 


94 


650281 


565 


349719 


5a 


6 


611012 


470 


960392 


94 


650620 


5(55 


349380 


54 


7 


611294 


470 


960335 


94 


650959 


564 


349041 


53 


8 


611576 


470 


960279 


94 


651297 


564 


348703 


52 


9 


611858 


469 


960222 


94 


651636 


564 


348364 


51 


10 


612140 


469 


960i 65 


94 


651974 


563 


348026 


50 


11 


9-612421 


469 


9-960109 


95 


9-652312 


563 


10-347688 


49 


12 


612702 


468 


960052 


95 


652650 


563 


347350 


48 


13 


612983 


468 


959995 


95 


652988 


563 


347012 


47 


14 


613264 


467 


959938 


95 


653326 


562 


346674 


46 


15 


613545 


467 


959882 


95 


653663 


562 


346337 


45 


16 


613825 


467 


959825 


95 


654000 


562 


346000 


44 


IV 


614 1 05 


466 


959768 


95 


654337 


561 


345663 


43 


18 


614385 


466 


959711 


95 


654674 


561 


345326 


42 


19 


614665 


466 


959654 


95 


655011 


561 


344989 


41 


20 


614944 


465 


959596 


95 


655348 


561 


344652 


40 


21 


9-615223 


465 


9-959539 


95 


9-655684 


560 


10-344316 


39 


22 


615502 


465 


959482 


95 


656020 


560 


343980 


38 


23 


615781 


464 


959425 


95 


656356 


560 


343644 


37 


24 


616060 


464 


959368 


95 


656692 


559 


343308 


36 


25 


616338 


464 


959310 


96 


657028 


559 


342972 


35 


26 


616616 


463 


959253 


96 


657364 


559 


342636 


34 


27 


616894 


463 


959195 


96 


657699 


559 


342301 


33 


28 


617172 


462 


959138 


96 


658034 


558 


341966 


32 


29 


617450 


462 


959081 


96 


658369 


558 


341631 


31 


30 


617727 


462 


959023 


96 


658704 


558 


341296 


30 


31 


9*618004 


461 


9-958965 


96 


9-659039 


558 


10-340961 


29 


32 


618281 


461 


958908 


96 


659373 


557 


340627 


28 


33 


618558 


461 


958850 


96 


659708 


557 


340292 


27 


34 


P18834 


460 


958792 


96 


660042 


557 


339958 


26 


35 


619110 


460 


958734 


96 


660376 


557 


339624 


25 


36 


619386 


460 


958677 


96 


660710 


556 


339290 


24 


37 


619662 


459 


958619 


96 


661043 


556 


338957 


23 


38 


619938 


459 


958561 


96 


661377 


556 


338623 


22 


39 


620213 


459 


958503 


97 


661710 


555 


338290 


21 


40 


620488 


458 


958445 


97 


662043 


555 


337957 


20 


41 


9-620763 


458 


9958387 


97 


9662376 


555 


10-337624 


19 


42 


621038 


457 


958329 


97 


662709 


554 


337291 


18 


43 


621313 


457 


958271 


97 


663042 


554 


336958 


17 


44 


621587 


457 


958213 


97 


663375 


554 


336625 


16 


45 


621861 


456 


958154 


97 


663707 


554 


336293 


15 


46 


622135 


456 


958096 


97 


664039 


553 


335961 


14 


47 


622409 


456 


958038 


97 


664371 


553 


335629 


13 


48 


622682 


455 


957979 


97 


664703 


553 


335297 


12 


49 


622956 


455 


957921 


97 


665035 


553 


334965 


11 


50 


623229 


455 


957863 


97 


665366 


552 


334634 


10 


51 


9-623502 


454 


9-957804 


97 


9-665697 


652 


1C-334303 


9 


52 


623774 


454 


957746 


98 


666029 


552 


333971 


8 


53 


624047 


454 


957687 


98 


666360 


551 


333640 


7 


54 


624319 


453 


957628 


98 


666691 


551 


333309 


6 


55 


624591 


453 


957570 


98 


667021 


551 


332979 


5 


56 


624863 


453 


957511 


98 


667352 


551 


332648 


4 


57 


625 J 35 


452 


957452 


98 


667682 


550 


332318 


3 


58 


625406 


452 


957393 


98 


668013 


550 


331987 


2 


59 


625677 


452 


957335 


98 


668343 


550 


331657 


1 


60 


625948 


451 


957276 


98 


668672 


550 


331328 






| Cosine I 



1 Sine 



| Cotang. | 



I Twff. | M 



65 Degrees. 



LOGARITHMIC SINES, COSINES, ETC. (25 Degrees.) 223 



M. | 



| Cosine I D. I Taug. | D. | Coiaiig. ! 






9625948 


451 


9-957276 


98 


9-668673 


550 


10-331327 1 


60 


J 


6262 J 9 


451 


957217 


98 


669002 


549 


330998 


59 


2 


626490 


451 


957158 


98 


669332 


549 


330668 


58 


3 


626760 


450 


957099 


98 


669661 


549 


330339 


57 


4 


627030 


450 


957040 


98 


669991 


548 


330009 


.56 


5 


627300 


450 


956981 


98 


670320 


548 


329680 


55 


6 


6275 70 


449 


956921 


99 


670649 


548 


329351 


54 


7 


627M0 


449 


956862 


99 


670977 


548 


329023 


53 


8 


628 i 09 


449 


956803 


99 


671306 


547 


328694 


52 


9 


628378 


448 


956744 


99 


671634 


547 


328366 


51 


10 


628G47 


448 


956684 


99 


671963 


547 


328037 


50 


11 


9 628916 


447 


9956625 


99 


9-672291 


547 


10-327709 


49 


12 


629)85 


447 


956566 


99 


672619 


546 


327381 


48 


13 


629453 


447 


956506 


99 


672947 


546 


327053 


47 


14 


629721 


446 


956447 


99 


673274 


546 


326726 


46 


15 


629989 


446 


956387 


99 


673602 


546 


326398 


45 


16 


630257 


446 


956327 


99* 


673929 


545 


326071 


44 


17 


630524 


446 


956268 


99 


6' 4257 


545 


325743 


4t 


18 


630792 


445 


956208 


100 


674584 


545 


325416 


42 


19 


631059 


445 


956148 


100 


674910 


544 


325090 


41 


20 


631326 


445 


956089 


100 


675237 


544 


324763 


40 


21 


9-831593 


444 


9-956029 


100 


9-675564 


544 


10-324436 


39 


22 


631859 


444 


955969 


100 


675*90 


544 


324110 


38 


23 


632425 


444 


955909 


100 


676216 


543 


323784 


37 


24 


632392 


443 


955849 


100 


676543 


543 


323457 


36 


25 


632658 


443 


955789 


100 


676869 


543 


323131 


35 


26 


632923 


443 


955729 


100 


677194 


543 


322806 


34 


27 


633 J 89 


442 


955669 


100 


677520 


542 


322480 


33 


28 


633454 


442 


955609 


100 


677846 


542 


322154 


32 


29 


633719 


442 


955548 


100 


678171 


542 


321829 


31 


30 


633984 


441 


955488 


100 


678496 


542 


321504 


30 


31 


9-634249 


441 


9-955428 


101 


9-678821 


541 


10-321179 


29 


32 


634514 


440 


955368 


101 


679146 


541 


320*54 


28 


33 


634778 


440 


955307 


101 


679471 


541 


320529 


27 


34 


635042 


440 


955247 


101 


679795 


541 


320205 


26 


35 


635306 


439 


955186 


101 


680120 


540 


319880 


25 


36 


635570 


439 


955126 


lill 


680444 


540 


319556 


24 


37 


635834 


439 


955065 


101 


680768 


540 


319232 


23 


38 


636097 


438 


955005 


101 


681092 


540 


318908 


22 


39 


636360 


438 


954944 


101 


681416 


539 


318584 


21 


40 


636623 


438 


954883 


101 


681740 


539 


318260 


20 


41 


9*636886 


437 


9-954823 


101 


9682063 


539 


10-317937 


19 


42 


637148 


437 


954762 


101 


682387 


539 


317613 


18 


43 


637411 


437 


954701 


101 


682710 


538 


317290 


17 


44 


637673 


437 


954640 


101 


683033 


538 


316967 


16 


45 


637935 


436 


954579 


101 


683356 


538 


316644 


15 


46 


638197 


436 


954518 


102 


683679 


538 


316321 


14 


47 


638458 


436 


954457 


102 


684001 


537 


315999 


13 


48 


638720 


435 


954396 


102 


684324 


537 


315676 


12 


49 


638981 


435 


954335 


102 


684646 


537 


315354 


11 


50 


639242 


435 


954274 


102 


684968 


537 


315032 


10 


51 


9 639503 


434 


9-954213 


102 


9-685290 


536 


10-314710 


9 


52 


639764 


434 


954152 


102 


685612 


536 


314388 


8 


53 


640024 


434 


954090 


102 


685934 


536 


314066 


7 


54 


640284 


433 


954029 


102 


686255 


536 


313745 


6 


55 


640544 


433 


953968 


102 


686577 


535 


313423 


5 


56 


640804 


433 


953906 


102 


686898 


535 


313102 


4 


57 


641064 


432 


953845 


102 


687219 


535 


312781 


3 


58 


641324 


432 


953783 


102 


687540 


535 


312460 


2 


59 


641584 


432 


953722 


103 


687861 


534 


312139 


1 


60 


1 641842 


431 


953660 


103 


688182 


534 


311818 







I Cosine 




| Sine 

64 


1 
Degre 


| Cotang. 
es. 


1 


1 Tang. 


| M. 



224 (20 Degrees.) LOGARITHMIC SINES, COS WES, ETC. 



M. 


Sine 


, D. 


Cosine 


D. 





9*641842 


431 


9-953660 


102 


1 


642J01 


431 


953599 


103 


2 


642360 


421 


953537 


103 


3 


642618 


430 


953475 


103 


4 


642877 


430 


953413 


103 


5 


643135 


430 


953352 


103 


6 


643393 


430 


953290 


103 


7 


643650 


429 


953228 


103 


8 


643908 


429 


953166 


103 


9 


644 1G5 


429 


953104 


103 


10 


644423 


428 


953042 


103 


11 


9-644680 


428 


9-952980 


104 


12 


644936 


428 


952918 


104 


13 


645193 


427 


952855 


104 


14 


645450 


427 


952793 


104 


15 


645706 


427 


952731 


104 


16 


645962 


.426 


952669 


104 


17 


646218 


426 


952606 


104 


18 


646474 


426 


952544 


104 


19 


646729 


425 


952481 


104 


SO 


646984 


425 


952419 


104 


21 


9647240 


425 


9-952356 


104 


22 


647494 


424 


952294 


104 


23 


647749 


424 


952231 


104 


24 


648004 


424 


952168 


105 


25 


648258 


424 


952106 


105 


26 


648512 


423 


952043 


105 


27 


648766 


423 


951980 


105 


28 


649020 


423 


951917 


105 


29 


649274 


422 


951854 


105 


30 


649527 


422 


951791 


105 


31 


9-649781 


422 


9-951728 


105 


32 


650034 


422 


951665 


105 


33 


650287 


421 


951602 


105 


34 


650539 


421 


951539 


105 


35 


650792 


421 


951476 


105 


36 


651044 


420 


951412 


105 


37 


651297 


420 


951349 


106 


38 


651549 


420 


951286 


106 


39 


651800 


419 


951222 


106 


40 


652052 


419 


951159 


106 


41 


9-652304 


419 


9-951096 


106 


42 


652555 


418 


951032 


106 


43 


652806 


418 


950968 


106 


44 


653057 


418 


950905 


106 


45 


653308 


418 


950841 


106 


46 


653558 


4.17 


950778 


106 


47 


653808 


417 


950714 


106 


48 


654059 


417 


950650 


106 


49 


654309 


416 


950586 


106 


50 


654558 


416 


950522 


107 


5] 


654808 


416 


9-950458 


107 


52 


655058 


416 


950394 


107 


53 


655307 


415 


950330 


107 


54 


655556 


415 


950266 


107 


55 


655805 


415 


950202 


107 


56 


656054 


414 


950138 


107 


57 


656302 


414 


950074 


107 


58 


656551 


414 


950010 


107 


59 


656799 


413 


949945 


107 


60 


657047 


413 


949881 


107 



Tang. 


D. | 


Cotang. 


1 


9-688182 


534 


10-311818 


60 


688502 


534 


311498 


59 


688823 


534 


311177 


58 


689143 


533 


310857 


57 


689463 


533 


310537 


56 


689783 


533 


310217 


55 


690103 


533 


309897 


54 


690423 


533 


309577 


53 


690742 


532 


309258 


52 


691062 


532 


308938 


51 


691381 


532 


308619 


50 


9-691700 


531 


10-308300 


49 


692019 


531 


307981 


48 


692338 


531 


307062 


47 


692656 


531 


307344 


46 


692975 


531 


307025 


45 


693293 


530 


306707 


44 


693612 


530 


306388 


43 


693930 


530 


306070 


42 


694248 


530 


305752 


41 


694566 


529 


305434 


40 


9-694883 


529 


10305117 


39 


695201 


529 


304799 


38 


695518 


529 


304482 


37 


695836 


529 


304164 


36 


696153 


528 


303847 


35 


696470 


528 


303530 


34 


696787 


528 . 


303213 


33 


697103 


528 


302897 


32 


697420 


527 


302580 


31 


697736 


527 


302264 


30 


9-698053 


527 


10-301947 


29 


698369 


527 


301631 


28 


698G85 


526 


301315 


27 


699001 


526 


300999 


2(5 


699316 


526 


300684 


25 


699632 


526 


300368 


24 


699947 


526 


300053 


23 


700263 


525 


299737 


22 


700578 


525 


299422 


21 


700893 


525 


299107 


20 


9-701208 


524 


10-298792 


19 


701523 


524 


298477 


18 


701837 


524 


298163 


17 


702152 


524 


297848 


16 


702466 


524 


297534 


15 


702780 


523 


297220 


14 


703095 


523 


296905 


13 


703409 


523 


296591 


12 


703723 


523 


296277 


11 


704036 


522 


295964 


10 


9 704350 


522 


10-295650 


9 


704663 


522 


295337 


8 


704977 


522 


295023 


7 


705290 


522 


294710 


6 


705603 


521 


294397 


5 


705916 


521 


294084 


4 


706228 


521 


293772 


3 


706541 


521 


293459 


2 


706854 


521 


293146 


1 


707166 


520 


292834 






| Cosine 1 



Sine | | Cotanp. | 

63 Degrees. 



J Tang. | 11 



LOGARITHMIC SIiVFS, COSWFS, ETC. (27 Degrees.) 225 



M. 


| Sine 


1 D- 


| Cosine 


1 D. 


Tang. 


D. 


Cotang. 







9-657047 


413 


9-949881 


107 


9-707166 


520 


10-292834 


60 


1 


657295 


413 


949816 


107 


707478 


520 


292522 


59 


2 


657542 


412 


949752 


107 


707790 


520 


292210 


58 


3 


657790 


412 


949688 


108 


708102 


520 


291898 


57 


4 


658037 


412 


949623 


108 


708414 


519 


291586 


56 


5 


658284 


412 


949558 


108 


708726 


519 


291274 


55 


6 


658531 


411 


949494 


108 


709037 


519 


290963 


54 


7 


658778 


411 


949429 


108 


709349 


519 


290651 


53 


8 


659025 


411 


949^64 


108 


709660 


519 


290340 


52 


9 


659271 


4)0 


949300 


108 


709971 


518 


290029 


51 


10 


6oyoi7 


4J0 


949235 


108 


710282 


518 


289718 


50 


11 


9-659763 


410 


9*949170 


108 


9-710593 


518 


10-289407 


49 


12 


660009 


4u9 


949105 


108 


710904 


518 


289096 


48 


13 


660255 


409 


949040 


108 


711215 


518 


288785 


47 


14 


660501 


409 


948975 


108 


711525 


517 


288475 


46 


15 


6607 46 


409 


948910 


108 


711836 


517 


288164 


45 


16 


660901 


408 


948845 


. 108 


712146 


517 


287854 


44 


17 


661236 


408 


948780 


109 


712456 


517 


287544 


43 


1< 


661481 


408 


948715 


109 


712766 


516 


287234 


42 


19 


661726 


407 


948650 


109 


713076 


516 


286924 


41 


20 


661970 


407 


948584 


109 


713386 


5i6 


286614 


40 


21 


9-662214 


407 


9-948519 


109 


9-713696 


516 


10-286304 


39 


22 


662459 


407 


948454 


109 


714005 


516 


285995 


38 


23 


662703 


406 


948388 


109 


714314 


515 


285686 


37 


24 


662946 


406 


948323 


109 


714624 


515 


285376 


36 


25 


663190 


406 


948257 


109 


714933 


515 


285067 


35 


26 


663433 


405 


948192 


109 


715242 


515 


284758 


34 


27 


6S3677 


405 


948126 


109 


715551 


511 


284449 


33 


28 


663920 


405 


948060 


109 


715860 


514 


284140 


32 


29 


664163 


405 


947995 


110 


716168 


514 


283832 


31 


30 


664406 


404 


947929 


110 


716477 


514 


283523 


30 


31 


9-664648 


404 


9-947863 


110 


9-716785 


514 


10-283215 


29 


32 


664891 


404 


947797 


110 


717093 


513 


282907 


28 


33 


665133 


403 


947731 


110 


717401 


513 


282599 


27 


34 


665375 


403 


947665 


110 


717709 


513 


282291 


26 


35 


665617 


403 


947600 


110 


718017 


513 


281983 


25 


36 


665859 


402 


947533 


-110 


718325 


513 


281675 


24 


37 


666100 


402 


947467 


110 


718633 


512 


281367 


23 


38 


666342 


402 


947401 


110 


718940 


512 


281060 


22 


39 


666583 


402 


947335 


110 


719248 


512 


280752 


21 


40 


666824 


401 


947269 


110 


719555 


512 


280445 


20 


41 


9-667065 


401 


9-947203 


110 


9-719862 


512 


10-280138 


19 


42 


667305 


401 


947136 


111 


720169 


511 


279831 


18 


43 


667546 


401 


947070 . 


111 


729476 


511 


279524 


17 


44 


667786 


400 


947004 


111 


720783 


511 


279217 


16 


45 


668027 


400 


946937 


111 


721089 


511 


278911 


15 


46 


668267 


400 


946871 


111 


721396 


511 


278604 


14 


47 


668506 


399 


946804 


111 


721702 


510 


278298 


13 


48 


668746 


399 


946738 


111 


722009 


510 


277991 


12 


49 


668986 


399 


946671 


111 


722315 


510 


277685 


11 


50 


669225 


399 


946604 


111 


722621 


510 


277379 


10 


51 


9-669464 


398 


9-946538 


111 


9-722927 


510 


10-277073 


9 


52 


669703 


398 


946471 


111 


723232 


509 


276768 


8 


53 


669942 


398 


946404 


111 


723538 


509 


276462 


7 


54 


670181 


397 


946337 


111 


723844 


509 


276156 


6 


55 


670419 


397 


946270 


112 


724149 


509 


275851 


5 


56 


670658 


397 


946203 


112 


724454 


509 


275546 


4 


57 


670896 


397 


946136 


112 


724759 


508 


275241 


3 


58 


671134 


396 


946069 


112 


725065 


508 


274935 


2 


59 


671372 


396 


946002 


112 


725369 


508 


274631 


1 


60 


671609 


396 


945935 


112 


725674 


508 


274326 






i CoHne ) 



| Sine | 



Cotang. | 



S Tang. I M. 



62 Degrees. 



226 (28 Degrees.) LOGARITHMIC SINES, COSINES, ETC. 



M. | 



| Cosine | 



Tang. 



D. | Cotang. | 



I Coaine 



Cotang. | 






9*671609 


396 


9-945935 


112 


9-725674 


508 


10-274326 


m 


1 


671847 


395 


945868 


IIS 


725979 


508 


274021 


59 


2 


672084 


395 


945800 


112 


726284 


507 


273716 


58 


3 


672321 


395 


945733 


112 


726588 


507 


273412 


:>7 


4 


672558 


395 


945666 


112 


726892 


507 


273108 


56 


5 


672795 


394 


945598 


112 


727197 


507 


272803 


55 


6 


673032 


394 


945531 


112 


727501 


507 


272499 


54 


7 


673268 


394 


945464 


113 


727805 


506 


272195 


53 


8 


673505 


394 


945396 


113 


728109 


506 


271891 


52 


9 


673741 


393 


945328 


113 


728412 


506 


271588 


51 


10 


673977 


393 


945261 


113 


728716 


506 


271284 


50 


11 


9674213 


393 


9-945193 


113 


9-729020 


506 


10-270980 


49; 


12 


674448 


392 


945125 


113 


729323 


505 


270677 


48 


13 


674684 


392 


945058 


113 


729626 


505 


270374 


47 


14 


674919 


392 


944990 


113 


729929 


505 


270071 


46 


15 


675155 


392 


944922 


113 


730233 


505 


269767 


45 


16 


675390 


391 


944854 


113 


. 730535 


505 


269465 


44 


17 


675624 


391 


944786 


113 


730838 


504 


269162 


43. 


18 


675859 


391 


944718 


113 


731141 


504 


268859 


42: 


19 


676094 


391 


944650 


113 


731444 


504 


268556 


41 


20 


676328 


390 


944582 


114 


731746 


504 


268254 


40 


SI 


9676562 


390 


9-944514 


114 


9-732048 


504 


10-267952 


3f 


22 


676796 


390 


944446 


114 


732351 


503 


267649 


38 


23 


677030 


390 


944377 


114 


732653 


503 


267347 


37 


24 


677264 


389 


944309 


114 


732955 


503 


267045 


3G 


£5 


677498 


389 


944241 


114 


733257 


503 


266743 


35 


26 


677731 


389 


944172 


114 


733558 


503 


266442 


34 


27 


677964 


388 


944104 


114 


733860 


502 


266140 


33 


28 


678197 


388 


944036 


114 


734162 


502 


265838 


32 


29 


678430 


388 


M3967 


114 


734463 


502 


265537 


31 


30 


678663 


388 


943899 


U4 


734764 


502 


265236 


30 


31 


9-678895 


387 


P943830 


114 


9-735066 


502 


10-264934 


29 


32 


679128 


387 


943761 


114 


735367 


502 


264633 


28 


33 


679360 


387 


943693 


115 


735668 


501 


264332 


27 


34 


679592 


387 


943624 


115 


735969 


501 


264031 


26 


35 


679824 


386 


943555 


115 


736269 


501 


263731 


25 


36 


680056 


386 


943486 


115 


736570 


501 


263430 


24 


37 


680288 


386 


943417 


115 


736871 


501 


263129 


23 


38 


680519 


385 


943348 


115 


737171 


500 


262829 


22 


39 


680750 


385 


943279 


115 


737471 


500 


262529 


21 


40 


680982 


385 


943210 


115 


737771 


500 


262229 


20 


41 


9-681213 


385 


9-943141 


115 


9-738071 


500 


10-261929 


19 


42 


681443 


384 


943072 


115 


438371 


500 


261629 


18 


43 


681674 


384 


943003 


115 


738671 


499 


261329 


17 


44 


681905 


384 


942934 


115 


738971 


499 


261029 


16 


45 


682135 


384 


942864 


115 


739271 


499 


260729 


15 


46 


682365 


383 


942795 


116 


739570 


499 


260430 


14 


47 


682595 


383 


942726 


116 


739870 


499 


260130 


13 


48 


682825 


383 


942656 


116 


740169 


499 


259831 


12 


49 


683055 


383 


942587 


116 


740468 


498 


259532 


11 


50 


683284 


382 


942517 


116 


740767 


498 


259233 


10 


51 


9-683514 


382 


9-942448 


116 


9-741066 


498 


10-258934 


9 


52 


683743 


382 


942378 


116 


741365 


498 


258635 


8 


53 


683972 


382 


942308 


116 


741664 


498 


258336 


7 


54 


684201 


381 


942239 


116 


741962 


497 


258038 


6 


55 


684430 


381 


942169 


116 


742261 


497 


257739 


5 


56 


684658 


381 


942099 


116 


742559 


497 


257441 


4 


57 


684887 


380 


942029 


116 


742858 


497 


257142 


3 


58 


685115 


380 


941959 


116 


743156 


497 


256844 


2 


59 


685343 


380 


941889 


117 


743454 


497 


256546 


1 


60 


685571 


380 


941819 


117 


743752 


496 


256248 






I Tang. | M 



61 Degrees. 



LOGARITHMIC SINES, COSINES, ETC. (29 Degrees.) 227 



M. 


Sine 


1 D- 


| Cosine 


I>. 


Tang. 


| D. 


| Cotang. 





9-685571 


380 


9-941819 


117 


9-743752 


496 


10256248 


1 


685799 


379 


941749 


117 


744050 


496 


255950 


2 


686027 


379 


941679 


117 


744348 


496 


255652 


3 


686254 


379 


941609 


117 


744645 


496 


255355 


4 


686482 


379 


941539 


117 


744943 


496 


255057 


5 


686709 


378 


941469 


117 


745240 


496 


254760 


6 


686936 


378 


941398 


117 


745538 


495 


254462 


7 


687163 


378 


941328 


117 


745835 


495 


254165 


8 


687389 


378 


941258 


117 


746132 


495 


253868 


9 


687616 


377 


941187 


117 


746429 


495 


253571 


10 


687843 


377 


941117 


117 


746726 


495 


253274 


n 


9-688069 


377 


9-941046 


118 


9-747023 


494 


10-252977 


12 


688295 


377 


940975 


118 


747319 


494 


252G81 


13 


688521 


376 


940905 


118 


747616 


494 


252384 


34 


688747 


376 


940834 


118 


747913 


494 


252087 


15 


688972 


376 


940763 


118 


748209 


494 


251791 


16 


689198 


376 


940693 


118 


748505 


493 


251495 


17 


689423 


375 


940622 


118 


748801 


493 


251199 


18 


689648 


375 


940551 


118 


749097 


493 


250903 


19 


689^73 


375 


940480 


118 


749393 


493 


250607 


20 


690098 


375 


940409 


118 


749689 


493 


250311 


21 


9-690323 


374 


9-940338 


118 


9-749985 


493 


10-250015 


22 


690548 


374 


940267 


118 


750281 


492 


249719 


23 


690772 


374 


940196 


118 


750576 


492 


249424 


24 


690996 


374 


940125 


119 


750872 


492 


249128 


25 


691220 


373 


940054 


119 


751167 


492 


248833 


26 


691444 


373 


939982 


119 


751462 


492 


248538 


27 


691668 


373 


939911 


119 


751757 


492 


248243 


28 


691892 


373 


939840 


119 


752052 


491 


247948 


29 


692115 


372 


939768 


119 


752347 


491 


247653 


30 


692339 


372 


939697 


119 


752642 


491 


247358 


31 


9-692562 


372 


9-939625 


119 


9-752937 


491 


10-247063 


32 


692785 


371 


939554 


119 


753231 


491 


246769 


33 


693008 


371 


939482 


119 


753526 


491 


246474 


34 


693231 


371 


939410 


119 


753820 


490 


246180 


35 


693453 


371 


939339 


119 


754115 


490 


245885 


36 


693676 


370 


939267 


120 


754409 


490 


245591 


37 


693898 


370 


939195 


120 


754703 


490 


245297 


38 


694120 


370 


939123 


120 


754997 


490 


245003 


39 


694342 


370 


939052 


120 


755291 


490 


244709 


40 


694564 


369 


938980 


120 


755585 


489 


244415 


41 


9-694786 


369 


9-938908 


120 


9-755878 


489 


10-244122 


42 


695007 


369 


938836 


120 


756172 


489 


243828 


43 


695229 


369 


938763 


120 


756465 


489 


243535 


44 


695450 


368 


938691 


120 


756759 


489 


243241 


45 


695671 


368 


938619 


120 


757052 


489 


242948 


46 


695892 


368 


938547 


120 


757345 


488 


242655 


47 


696113 


368 


938475 


120 


757638 


488 


242362 


48 


696334 


367 


938102 


121 


757931 


488 


242069 


49 


696554 


367 


938330 


121 


758224 


488 


241776 


50 


696775 


367 


938258 


121 


758517 


488 


241483 


51 


9-696995 


367 


9-938185 


121 


9-758810 


488 


10-241190 


52 


697215 


366 


938113 


121 


759102 


487 


240898 


53 


697435 


366 


938040 


121 


759395 


487 


240605 


54 


697654 


366 


937967 


121 


759687 


487 


240313 


55 


697874 


366 


937895 


121 


• 759979 


487 


240021 


56 


698094 


365 


937822 


121 


760272 


487 


239728 


57 


i 698313 


365 


937749 


121 


760564 


487 


239436 


58 


698532 


365 


937676 


121 


760856 


486 


239144 


59 


698751 


365 


937604 


121 


761148 


486 


238852 


60 


698970 


364 


937531 


121 


761439 


486 


238561 



J Cosine \ 



| 1 Cotang. 

60 Degrees. 



1 Tang. | M. 



228 (30 Degrees.) LOGARITHMIC SINES, COSINES, ETC. 



M. 


| Sine 


1 D. 


Cosine 


1 D- 


1 Tang. 


1 D. 


( Cotang. 


I 





9-698970 


364 


9937531 


121 


9-761439 


486 


1(1-238561 


60 


1 


699189 


364 


937458 


122 


761731 


486 


238269 


59 


2 


699407 


364 


937385 


122 


762023 


486 


237977 


58 


3 


699626 


364 


937312 


122 


762314 


486 


237686 


57 


4 


699844 


363 


937238 


122 


762606 


485 


237394 


56 


5 


700062 


363 


937165 


122 


762897 


485 


237103 


55 


6 


700280 


363 


937092 


122 


763188 


485 


236812 


54 


7 


700498 


363 


937019 


122 


763479 


485 


236521 


53 


8 


700716 


363 


936946 


122 


763770 


485 


236230 


JQ 


9 


700933 


362 


936872 


122 


764061 


485 


235939 


51 


30 


701151 


362 


936799 


122 


764352 


484 


235648 


50 


11 


9-701368 


362 


9936725 


122 


9-764643 


484 


10-235357 


49 


12 


701585 


362 


936652 


123 


764933 


484 


235067 


48 


13 


701802 


361 


936578 


123 


765224 


484 


234776 


47 


14 


702019 


361 


936505 


123 


765514 


484 


234486 


46 


15 


702236 


361 


936431 


123 


765805 


484 


234195 


45 


16 


702452 


361 


936357 


123 


766095 


484 


233905 


44 


17 


702669 


360 


936284 


123 


766385 


483 


233615 


43 


18 


702885 


360 


936210 


1*3 


766675 


483 


233325 


42 


19 


703101 


360 


936136 


123 


766965 


483 


233035 


41 


20 


703317 


360 


936062 


123 


767255 


483 


232745 


40 


21 


9-703533 


359 


9-935988 


123 


9-767545 


483 


10-232455 


39 


22 


703749 


359 


935914 


123 


767834 


483 


232166 


38 


23 


703964 


359 


935840 


123 


768124 


482 


231876 


37 


24 


7041/9 


359 


935766 


124 


768413 


482 


231587 


36 


25 


704395 


359 


935692 


124 


768703 


482 


231297 


35 


26 


704610 


358 


935618 


124 


768992 


482 


231008 


34 


27 


704825 


358 


935543 


124 


769281 


482 


230719 


33 


28 


705040 


358 


935469 


124 


769570 


482 


230430 


32 


29 


705254 


358 


935395 


124 


769860 


481 


230140 


31 


30 


705469 


357 


935320 


124 


770148 


481 


229852 


30 


31 


9-705683 


357 


9-935246 


124 


9-770437 


481 


10-229563 


29 


32 


705898 


357 


935171 


124 


770726 


481 


229274 


28 


33 


706112 


357 


935097 


124 


771015 


481 


228985 


27 


34 


706326 


356 


935022 


124 


771303 


481 


228697 


26 


35 


706539 


356 


934948 


124 


771592 


481 


228408 


25 


36 


706753 


356 


934873 


124 


771880 


480 


228120 


24 


37 


706967 


356 


934798 


125 


772168 


480 


227832 


23 


38 


707180 


355 


934723 


125 


772457 


480 


227543 


22 


39 


707393 


355 


934649 


125 


772745 


480 


227255 


24 


40 


707606 


355 


934574 


125 


773033 


480 


226967 


20 


41 


9-707819 


355 


9934499 


125 


9-773321 


480 


10-226679 


19 


42 


708032 


354 


934424 


125 


773608 


479 


226392 


18 


43 


708245 


354 


934349 


125 


773896 


479 


226104 


17 


44 


708458 


354 


934274 


125 


774184 


479 


225816 


16 


45 


708670 


354 


934199 


125 


774471 


479 


225529 


15 


46 


708882 


353 


934123 


125 


774759 


479 


225241 


14 


47 


709094 


353 


934048 


125 


775046 


479 


224954 


13 


48 


709306 


353 


933973 


125 


775333 


479 


224667 


12 


49 


709518 


353 


933898 


126 


775621 


478 


224379 


11 


50 


709730 


353 


933822 


126 


775908 


478 


224092 


10 


51 


9-709941 


352 


9-933747 


126 


9-776195 


478 


10-223805 


9 


52 


710153 


352 


933671 


126 


776482 


478 


223518 


8 


53 


710364 


352 


933596 


126 


776769 


478 


223231 


7 


54 


710575 


352 


933520 


126 


777055 


478 


222945 


6 


55 


710786 


351 


933445 . 


126 


777342 


478 


222658 


5 


56 


710997 


351 


933369 


126 


777628 


477 


222372 


4 


57 


711208 


351 


933293 


126 


777915 


477 


222085 


3 


58 


711419 


351 


933217 


126 


778201 


477 


221799 


2 


59 


711629 


350 


933141 


126 


778487 


477 


221512 


1 


GO 


711839 


350 


933066 


126 


778774 


477 


221226 






I Cosine | 



Sine | | Cotang. | 

59 Degrees. 



| Tang. J M 



LOGARITHMIC SINES, COSINES, ETC. (31 Degrees.) 229 



M. 


Sine 


D. 


Cosine 


D. | 


Tfng. | 


D. | 


Cotangv 1 







9711839 


350 


9-933066 


126 


9-778774 


477 


10-221226 


60 


1 


712050 


350 


932990 


127 


779060 


477 


220940 


59 


2 


712260 


350 


932914 


127 


779346 


476 


220654 


58 


3 


712469 


349 


932838 


127 


779632 


476 


220368 


57 


4 


712679 


349 


932762 


127 


779918 


476 


220082 


56 


5 


712889 


349 


932685 


127 


780203 


476 


219797 


55 


6 


713098 


349 


932609 


127 


780489 


476 


219511 


54 


7 


713308 


349 


932533 


127 


780775 


476 


219225 


53 


8 


713517 


348 


932457 


127 


781060 


476 


218940 


52 


9 


713726 


348 


932380 


127 


781 340 


475 


218654 


51 


10 


713935 


348 


932304 


127 


781631 


475 


218369 


50 


11 


9714144 


348 


9-932228 


127 


9-781916 


475 


10-218084 


49 


12 


714352 


347 


932151 


127 


782201 


475 


217799 


48 


13 


714561 


347 


932075 


128 


782486 


475 


217514 


47 


14 


714769 


347 


931998 


128 


782771 


475 


217229 


46 


15 


714978 


347 


931921 


128 


783056 


475 


216944 


45 


16 


715186 


347 


931845 


128 


783341 


475 


216659 


44 


17 


715394 


346 


931768 


128 


783626 


474 


2J6374 


43 


18 


715602 


346 


931691 


128 


783910 


474 


216090 


42 


19 


715809 


346 


931614 


128 


784195 


474 


215805 


41 


20 


716017 


346 


931537 


128 


784479 


474 


21552J 


40 


21 


9716224 


345 


9931460 


128 


9-784764 


474 


10-2J5236 


39 


22 


716432 


345 


931383 


128 


785048 


474 


214952 


38 


23 


716639 


345 


931306 


128 


785332 


473 


214668 


37 


24 


716846 


345 


931229 


129 


785616 


473 


214384 


36 


25 


717053 


345 


931152 


129 


785900 


473 


214100 


35 


26 


717259 


344 


931075 


129 


786184 


473 


213816 


34 


27 


717466 


344 


930998 


129 


786468 


473 


213532 


33 


28 


717673 


344 


930921 


129 


786752 


473 


213248 


32 


29 


717879 


344 


930843 


129 


787036 


473 


212964 


31 


30 


718085 


343 


930766 


129 


787319 


472 


212681 


30 


31 


9-718291 


343 


9-930688 


129 


9-787603 


472 


10-212397 


29 


32 


718497 


343 


93061 1 


129 


787886 


472 


212114 


28 


33 


718703 


343 


930533 


129 


788170 


472 


211830 


27 


34 


718909 


343 


930456 


129 


788453 


472 


211547 


26 


35 


719114 


342 


930378 


129 


788736 


472 


211264 


25 


36 


719320 


342 


930300 


130 


789019 


472 


210981 


24 


37 


719525 


342 


930223 


130 


789302 


471 


210698 


23 


38 


719730 


342 


930145 


130 


789585 


471 


210415 


22 


39 


719935 


341 


930067 


130 


789868 


471 


210132 


21 


40 


720140 


341 


929989 


130 


790151 


471 


209849 


20 


41 


9-720345 


341 


9-929911 


130 


9-790433 


471 


10-209567 


19 


42 


720549 


341 


929833 


130 


790716 


471 


209284 


18 


43 


720754 


340 


929755 


130 


790999 


471 


209001 


17 


44 


720958 


340 


929677 


130 


791281 


471 


208719 


16 


45 


721162 


340 


929599 


130 


791563 


470 


208437 


15 


46 


721366 


340 


929521 


130 


791846 


470 


208154 


14 


47 


721570 


340 


929442 


130 


792128 


470 


207872 


13 


48 


721774 


339 


929364 


131 


792410 


470 


207590 


12 


49 


721978 


339 


929286 


131 


792692 


470 


207308 


11 


50 


722181 


339 


929207 


131 


792974 


470 


207026. 


10 


51 


9-722385 


339 


9-929129 


131 


9-793256 


470 


10-206744 


9 


52 


722588 


339 


929050 


131 


793538 


469 


206462 


8 


53 


722791 


338 


928972 


131 


793819 


469 


206181 


7 


54 


722994 


338 


928893 


131 


794101 


469 


205899 


6 


55 


723197 


338 


928815 


131 


794383 


469 


205617 


5 


56 


723400 


338 


928736 


131 


794664 


469 


205336 


4 


57 


723603 


337 


928657 


131 


794945 


469 


205055 


3 


58 


723805 


337 


928578 


131 


795227 


469 


204773 


2 


59 


724007 


337 


928499 


131 


795508 


468 


204492 


1 


60 


724210 


337 


928420 


131 


795789 


468 


204211 


• 



| Cosine J 



1 | Cotang. | 

58 Degrees. 



J Tan;. J M. 



230 (32 Degrees.) LOGARITHMIC SINES, COSINES, ETC. 



M. | Sine | D. \ Cosine 1 



| Tang. | D. \ Cotang. 



| Cosine | 



Sine | 



Cotang. | 






9724210 


337 


9-928420 


132 


9-795789 


468 


10-204211 


60 


1 


724412 


337 


928342 


132 


796070 


468 


203930 


59 


2 


724614 


336 


928263 


132 


796351 


468 


203649 


58 


3 


724816 


336 


928183 


132 


796632 


468 


203368 


57 


4 


725017 


336 


928104 


132 


796913 


468 


203087 


56 


5 


725219 


336 


928025 


132 


797194 


468 


202806 


55 


6 


725420 


335 


927946 


132 


797475 


468 


202525 


54 


7 


725622 


335 


927867 


132 


797755 


468 


202245 


53 


8 


725823 


335 


927787 


132 


798036 


467 


201964 


52 


9 


726024 


335 


927708 


132 


798316 


467 


201684 


51 


10 


726225 


335 


927629 


132 


798596 


467 


201404 


50 


11 


9-726426 


334 


9-927549 


132 


9-798877 


467 


10-201123 


\ 49 
48 


12 


726626 


334 


927470 


133 


799157 


467 


200843 


13 


726827 


334 


927390 


133 


799437 


467 


200563 


47 


14 


727027 


334 


927310 


133 


799717 


467 


200283 


46 


15 


727228 


334 


927231 


133 


799997 


466 


200003 


45 


16 


727428 


333 


927151 


133 


800277 


466 


199723 


44 


17 


727628 


333 


927071 


133 


800557 


466 


199443 


43 


18 


727828 


333 


926991 


133 


800836 


466 


199164 


42 


19 


728027 


333 


926911 


133 


801116 


466 


198884 


41 


20 


728227 


333 


926831 


133 


801396 


466 


198604 


40 


21 


9728427 


332 


9-926751 


133 


9-801675 


466 


10*198325 


39 


22 


728626 


332 


926671 


133 


801955 


466 


198045 


38 


23 


728825 


332 


926591 


133 


802234 


465 


197766 


37 


24 


729024 


332 


926511 


134 


802513 


465 


197487 


36 


25 


729223 


331 


926431 


134 


802792 


465 


197208 


35 


26 


729422 


331 


926351 


134 


803072 


465 


196928 


34 


27 


729621 


331 


926270 


134 


803351 


465 


196649 


33 


28 


729820 


331 


926190 


134 


803630 


465 


196370 


32 


29 


730018 


330 


926110 


134 


803908 


465 


196092 


31 


30 


730216 


330 


926029 


134 


804187 


465 


195813 


30 


31 


9730415 


330 


9-925949 


134 


9-804466 


464 


10-195534 


29 


32 


730613 


330 


925868 


134 


804745 


464 


195255 


28 


33 


730811 


330 


925788 


134 


805023 


464 


194977 


27 


34 


731009 


329 


925707 


134 


805302 


464 


194698 


26 


35 


731206 


329 


925626 


134 


805580 


464 


194420 


25 


36 


731404 


329 


925545 


135 


805859 


464 


194141 


24 


37 


731602 


329 


925465 


135 


806137 


464 


193863 


23 


38 


731799 


329 


925384 


135 


806415 


463 


193585 


22 


39 


731996 


328 


925303 


135 


806693 


463 


193307 


21 


40 


732193 


328 


925222 


135 


806971 


463 


193029 


20 


41 


9-732390 


328 


9-925141 


135 


9-807249 


463 


10192751 


19 


42 


732587 


328 


925060 


135 


807527 


463 


192473 


18 


43 


732784 


328 


924979 


135 


807805 


463 


192195 


17 


44 


732980 


327 


924897 


135 


808083 


463 


191917 


16 


45 


733177 


327 


924816 


135 


808361 


463 


191639 


15 


40 


733373 


327 


924735 


136 


808638 


462 


191362 


14 


47 


733569 


327 


924654 


136 


808916 


462 


191084 


13 


48 


733765 


327 


924572 


136 


809193 


462 


190807 


12 


49 


733961 


326 


924491 


136 


809471 


462 


190529 


11 


50 


734157 


326 


924409 


136 


809748 


462 


190252 


10 


51 


9734353 


326 


9-924328 


136 


9-810025 


462 


10-189975 


9 


52 


734549 


326 


924246 


136 


810302 


462 


189698 


8 


53 


734744 


325 


924 104 


136 


813580 


462 


189420 


7 


54 


734939 


325 


924083 


136 


810857 


462 


189143 


6 


55 


735135 


325 


924001 


136 


811134 


461 


186866 


5 


56 


735330 


325 


923919 


136 


811410 


461 


188590 


4 


57 


735525 


325 


923837 


136 


811687 


461 


188313 


3 


58 


735719 


324 


923755 


137 


811964 


461 


188036 


2 


59 


735914 


324 


923673 


137 


812241 


461 


187759 


1 


00 


736109 


324 


1 923591 


137 


812517 


461 


187483 






| Tang. f M. 



57 Degrees. 



LOGARITHMIC SINES, COSINES, ETC. (33 Degrees.) 231 



M. \ 


Sine ! 


D. | 


Cosine | 


D. | 


Tang. 


D. 


Cotang. ' 







9-736109 


324 


9-923591 


137 


9-812517 


461 


10-187482 


00 


1 


736303 


324 


923509 


137 


812794 


461 


187206 


59 


2 


736498 


324 


923427 


137 


813070 


461 


186930 


58 


3 


736692 


323 


923345 


137 


813347 


460 


186653 


57 


4 


736886 


323 


923263 


137 


813623 


460 


186377 


56 


5 


737080 


323 


923181 


137 


813899 


460 


186101 


55 


6 


737274 


323 


923098 


137 


814175 


460 


185825 


54 


7 


737467 


323 


923016 


137 


814452 


460 


185548 


53 


8 


737661 


322 


922933 


137 


814728 


460 


185272 


52 


9 


737855 


322 


922851 


137 


815004 


460 


184996 


51 


10 


738048 


322 


922768 


138 


815279 


460 


184721 


50 


11 


9-738241 


322 


9-922686 


138 


9-815555 


459 


10184445 


49 


12 


738434 


322 


922603 


138 


815831 


459 


184169 


48 


13 


738627 


321 


922520 


138 


816107 


459 


183893 


47 


14 


738820 


321 


922438 


138 


816382 


459 


183618 


46 


15 


739013 


321 


922355 


138 


816658 


459 


183342 


45 


16 


739206 


321 


922272 


138 


816933 


459 


183067 


44 


17 


739398 


321 


922189 


138 


817209 


459 


182791 


43 


18 


739590 


320 


922106 


138 


817484 


459 


182516 


42 


19 


739783 


320 


922023 


138 


817759 


459 


182241 


41 


20 


739975 


320 


921940 


138 


818035 


458 


181965 


40 


21 


9-740167 


320 


9-921857 


139 


9-818310 


458 


10-181690 


39 


22 


740359 


320 


921774 


139 


818585 


458 


181415 


38 


23 


740550 


319 


921691 


139 


818860 


458 


181140 


37 


24 


740742 


319 


921607 


139 


819135 


458 


180865 


36 


25 


740934 


319 


921524 


139 


819410 


458 


180590 


35 


26 


741125 


319 


921441 


139 


819684 


458 


180316 


34 


27 


741316 


319 


921357 


139 


819959 


458 


180041 


33 


28 


741508 


318 


921274 


139 


820234 


458 


179766 


32 


29 


741699 


318 


921190 


139 


820508 


457 


179492 


31 


30 


74 J 889 


318 


921107 


139 


820783 


457 


179217 


30 


31 


9-742080 


318 


9-921023 


139 


9-821057 


457 


10-178943 


29 


32 


742271 


318 


920939 


140 


821332 


457 


178668 


28 


33 


742462 


317 


920856 


140 


821606 


457 


178394 


27 


34 


742652 


317 


920772 


140 


821880 


457 


178120 


26 


35 


742842 


317 


920688 


140 


822154 


457 


177846 


25 


36 


743033 


317 


920604 


140- 


822429 


457 


177571 


24 


37 


743223 


317 


920520 


140 


822703 


457 


177297 


23 


38 


743413 


316 


920436 


140 


822977 


456 


177023 


22 


39 


743602 


316 


920352 


140 


823250 


456 


176750 


21 


40 


743792 


316 


920268 


140 


823524 


456 


176476 


20 


41 


9-743982 


316 


9-920184 


140 


9-823798 


456 


10-176202 


19 


42 


744171 


316 


920099 


140 


824072 


456 


175928 


18 


43 


744361 


315 


920015 


140 


824345 


456 


175655 


17 


44 


744550 


315 


919931 


141 


824619 


456 


175381 


16 


45 


744739 


315 


919846 


141 


824893 


456 


175107 


15 


46 


744928 


315 


919762 


141 


825166 


456 


174834 


14 


47 


745117 


315 


919677 


141 


825439 


455 


174561 


13 


48 


745306 


314 


919593 


141 


825713 


455 


174287 


12 


49 


745494 


314 


919508 


141 


825986 


455 


174014 


11 


50 


745683 


314 


919424 


141 


826259 


455 


173741 


10 


51 


9-745871 


314 


9-919339 


141 


9-826532 


455 


10173468 


9 


52 


746059 


314 


919254 


141 


826805 


455 


173195 


8 


53 


746248 


313 


919169 


141 


827078 


455 


172922 


7 


54 


746436 


313 


919085 


141 


827351 


455 


172649 


6 


55 


746624 


313 


919000 


141 


827624 


455 


172376 


5 


56 


746812 


313 


918915 


142 


827897 


454 


172103 


4 


57 


746999 


313 


918830 


142 


828170 


454 


171830 


3 


58 


747187 


312 


918745 


142 


828442 


454 


171558 


2 


59 


747374 


312 


918659 


142 


828715 


454 


171285 


i 1 


60 


747562 


312 


918574 


142 


828987 


454 


171013 


1 



| Cosine I 



Sine | 



| Cotang. | 



Tang. I K 



56 Degrees, 



232 (34 Degrees.) LOGARITHMIC SINES, COSINES, ETC. 



M. 


| Sine 


1 D. 


| Cosine 


1 D. 


Targf. 


\ D. 


Cotang. 







9-747562 


312 


9-918574 


142 


9-828987 


454 


10171013 


60 


1 


747749 


312 


918489 


142 


829260 


454 


170740 


59 


2 


747936 


312 


918404 


142 


829532 


454 


170468 


58 


3 


748123 


311 


918318 


142 


829805 


454 


170195 


57 


4 


748310 


311 


918233 


142 


830077 


454 


169923 


56 


5 


748497 


311 


918147 


142 


830349 


453 


169651 


55 


6 


748683 


311 


918062 


142 


830621 


453 


169379 


54 


7 


748870 


311 


917976 


143 


880893 


453 


169107 


53 


8 


749056 


310 


917891 


143 


831165 


453 


168835 


52 


9 


749243 


310 


917805 


143 


831437 


453 


168563 


51 


10 


749429 


310 


917719 


143 


831709 


453 


168291 


50 


11 


9-749615 


310 


9-917634 


143 


9-831981 


453 


10-168019 


49 


12 


749801 


310 


917548 


143 


832253 


453 


167747 


48 


13 


749987 


309 


917462 


143 


832525 


453 


167475 


47 


14 


750172 


309 


917376 


143 


832796 


453 


167204 


46 


15 


750358 


309 


917290 


143 


833068 


452 


166932 


45 


16 


750543 


309 


917204 


143 


833339 


452 


166661 


44 


17 


750729 


309 


917118 


144 


833611 


452 


166389 


43 


18 


750914 


308 


917032 


144 


833882 


452 


166118 


42 


19 


751099 


308 


916946 


144 


834154 


452 


165846 


41 


20 


751284 


308 


916859 


144 


834425 


452 


165575 


40 


21 


9751469 


308 


9-916773 


144 


9-834696 


452 


10165304 


39 


22 


751654 


308 


916687 


144 


834967 


452 


165033 


38 


23 


751839 


308 


916600 


144 


835238 


452 


1(54762 


37 


24 


752023 


307 


916514 


144 


835509 


452 


164491 


36 


25 


752208 


307 


916427 


144 


835780 


451 


164220 


35 


26 


752392 


307 


916341 


144 


• 836051 


451 


163949 


34 


27 


752576 


307 


916254 


144 


836322 


451 


163678 


33 


28 


752760 


307 


916167 


145 


836593 


451 


163407 


32 


29 


752944 


306 


916081 


145 


836864 


451 


163136 


31 


30 


753128 


306 


915994 


145 


837134 


451 


162866 


30 


31 


9753312 


306 


9-915907 


145 


9837405 


451 


10162595 


29 


32 


753495 


306 


915820 


145 


837675 


451 


162325 


28 


33 


753679 


306 


915733 


145 


837946 


451 


162054 


27 


34 


753862 


305 


915646 


145 


838216 


451 


1G1784 


20 


35 


754046 


305 


915559 


145 


838487 


450 


161513 


25 


36 


754229 


305 


915472 


145 


838757 


450 


161243 


24 


37 


754412 


305 


915385 


145 


839027 


450 


160973 


23 


38 


754595 


305 


915297 


145 


839297 


450 


160703 


22 


39 


754778 


304 


915210 


145 


839568 


450 


160432 


21 


40 


754960 


304 


915123 


146 


839838 


450 


160162 


20 


41 


9-755143 


304 


9-915035 


146 


9-840108 


450 


10159892 


19 


42 


755326 


304 


914948 


146 


840378 


450 


159622 


18 


43 


755508 


304 


914860 


146 


840647 


450 


159353 


17 


44 


755690 


304 


914773 


146 


840917 


449 


159083 


16 


45 


755872 


303 


914685 


146 


841187 


449 


158813 


15 


46 


756054 


303 


914598 


146 


841457 


449 


158543 


14 


47 


756236 


303 


914510 


146 


841726 


449 


158274 


13 


48 


756418 


303 


914422 


146 


841996 


449 


158004 


12 


49 


756600 


303 


914334 


146 


842266 


449 


157734 


11 


50 


756782 


302 


914246 


147 


842535 


449 


157465 


10 


51 


9-756963 


302 


9-914158 


147 


9-842805 


449 


10157195 


9 


52 


757144 


302 


914070 


147 


843074 


449 


156926 


8 


53 


757326 


302 


913982 


147 


843343 


449 


156657 


7 


54 


757507 


302 


913894 


147 


843612 


449 


156388 


6 


55 


757688 


301 


913806 


147 


843882 


448 


156118 


5 


56 


757869 


301 


913718 


147 


844151 


448 


155849 


4 


57 


758050 


301 


913630 


147 


844420 


448 


155580 


3 


58 


758230 


301 


913541 


147 


844689 


448 


15531 1 


2 


59 


758411 


301 


913453 


147 


844958 


448 


155042 


1 


60 


758591 


301 


913365 


147 


845227 


448 


154773 






| Cosine | 



I 



t Cotang. | 



I Tang. | M. 



55 Degrees. 



LOGARITHMIC SINES, COSINES, ETC. (35 Degrees.) 233 



| D. | Cosine \ D. | Tang. \ D. | Cotang. 






9-758591 


301 


9-913365 


147 


9.845227 


448 


10-154773 


60 


1 


758772 


300 


913276 


147 


845496 


448 


154504 


59 


o 


758952 


300 


913187 


148 


845764 


448 


154236 


58 


3 


759132 


300 


913099 


148 


846033 


448 


153967 


57 


4 


759312 


300 


913010 


148 


846302 


448 


153698 


56 


5 


759492 


300 


912922 


148 


846570 


447 


153430 


55 


6 


759672 


299 


912833 


148 


846839 


447 


153161 


54 


7 


759852 


299 


912744 


148 


847107 


447 


152893 


53 


8 


760031 


299 


912655 


148 


847376 


447 


152624 


52 


9 


760211 


299 


912566 


148 


847644 


447 


152356 


51 


10 


760390 


299 


912477 


148 


847913 


447 


152087 


50 


11 


9-760569 


298 


9-912388 


148 


9.848181 


447 


10151819 


49 


12 


760748 


298 


912299 


149 


848449 


447 


151551 


48 


13 


760927 


298 


912210 


149 


848717 


447 


151283 


47 


14 


761106 


298 


912121 


149 


848986 


447 


151014 


46 


15 


761285 


298 


912031 


149 


849254 


447 


150746 


45 


16 


761464 


298 


911942 


149 


849522 


447 


150478 


44 


17 


761642 


297 


911853 


149 


849790 


446 


150210 


43 


18 


761821 


297 


911763 


149 


850058 


446 


149942 


42 


19 


761999 


297 


911674 


149 


850325 


446 


149675 


41 


20 


762177 


297 


911584 


149 


850593 


446 


149407 


40 


21 


9762356 


297 


9-911495 


149 


9850861 


446 


10149139 


39 


22 


762534 


296 


911405 


149 


851129 


446 


148871 


38 


23 


762712 


296 


911315 


150 


851396 


446 


148604 


37 


24 


762889 


296 


911226 


150 


851664 


446 


148336 


36 


25 


763067 


296 


911136 


150 


851931 


446 


148069 


35 


26 


763245 


296 


911046 


150 


852199 


446 


147801 


34 


27 


763422 


296 


910956 


150 


852466 


446 


147534 


33 


28 


763600 


295 


910866 


150 


852733 


445 


147267 


32 


29 


763777 


295 


910776 


150 


853001 


445 


146999 


31 


30 


763954 


295 


910686 


150 


853268 


445 


146732 


30 


31 


9-764131 


295 


9-910596 


150 


9-853535 


445 


10-146465 


29 


32 


764308 


295 


910506 


150 


853802 


445 


146198 


28 


33 


764485 


294 


910415 


150 


854069 


445 


145931 


27 


34 


764662 


294 


910325 


151 


854336 


445 


145664 


26 


35 


764838 


294 


910235 


151 


854603 


445 


145397 


25 


36 


765015 
765191 


294 

294 


910144 
910054 


15L 


854870 


445 
445 


145130 
144863 


24 


37 


151 


855137 


23 


38 


765367 


294 


909963 


151 


855404 


445 


144596 


22 


39 


765544 


293 


909873 


151 


855671 


444 


144329 


21 


40 


765720 


293 


909782 


151 


855938 


444 


144062 


20 


41 


9-765896 


293 


9-909691 


151 


9-856204 


444 


10143796 


19 


42 


766072 


293 


909601 


151 


856471 


444 


143529 


18 


43 


766247 


293 


909510 


151 


856737 


444 


143263 


17 


44 


766423 


293 


909419 


151 


857004 


444 


142996 


16 


45 


766598 


292 


909328 


152 


857270 


444 


142730 


15 


46 


766774 


292 


909237 


152 


857537 


444 


142463 


14 


47 


766949 


292 


909146 


152 


857803 


444 


142197 


13 


48 


767124 


292 


909055 


152 


858069 


444 


141931 


12 


49 


767300 


292 


908964 


152 


858336 


444 


141664 


11 


50 


787475 


291 


908873 


152 


858602 


443 


141398 


10 


51 


9-767649 


291 


9-908781 


152 


9-858863 


443 


10-1411H2 


9 


52 


767824 


291 


908690 


152 


859134 


443 


140866 


8 


53 


767999 


291 


908599 


152 


859400 


443 


140600 


7 


54 


768173 


291 


903507 


152 


859666 


443 


140334 


6 


55 


768348 


290 


908416 


153 


859932 


443 


140068 


5 


56 


768522 


290 


908324 


153 


860198 


443 


139802 


4 


57 


768697 


290 


908233 


153 


860464 


443 


139536 


3 


58 


768871 


290 


908141 


153 


860730 


443 


139270 


2 


59 


769045 


290 


908049 


153 


860995 


443 


139005 


1 


00 


769219 


290 


907958 


153 


861261 


443 


138739 






t Cosine ) 



| Cotang. | 



| Tang. i M. 



54 Degrees. 



234 (36 Degrees.) LOGARITHMIC SINES, COSINES, ETC. 



W. | 



| Cosine | D. \ Tang. | D. | Cotang. 






9769219 


290 


9907958 


153 


9-861261 


443 


10138739 


m 


1 


769393 


289 


907866 


153 


861527 


443 


138473 


59 


2 


769566 


289 


907774 


153 


861792 


442 


138208 


53 


3 


769740 


289 


907682 


153 


862058 


442 


137942 


57 


4 


769913 


289 


907590 


153 


862323 


44£ 


137677 


56 


5 


770087 


289 


907498 


153 


862589 


442 


137411 


55 


6 


770260 


288 


907406 


153 


862854 


442 


137146 


.54 


7 


770433 


288 


907314 


154 


863119 


442 


136881 


53 


8 


770606 


288 


907222 


154 


863385 


442 


136615 


52 


9 


770779 


288 


907129 


154 


863650 


442 


136350 


51 


10 


770952 


288 


907037 


154 


863915 


442 


136085 


50 


11 


9-77^125 


288 


9-906945 


154 


9-864180 


442 


10135820 


49 


12 


771298 


287 


906852 


154 


864445 


442 


135555 


48 


13 


771470 


287 


906760 


154 


864710 


442 


135290 


47 


14 


771643 


287 


906667 


154 


864975 


441 


135025 


46 


1«> 


771815 


287 


906575 


154 


865240 


441 


134760 


45 


16 


771987 


287 


906482 


154 


865505 


441 


134495 


44 


17 


772159 


287 


906389 


155 


865770 


441 


134230 


43 


18 


772331 


286 


906296 


155 


866035 


441 


133965 


42 


19 


772503 


286 


906204 


155 


866300 


441 


133700 


41 


20 


772675 


286 


906111 


155 


866564 


441 


133436 


40 


21 


9-772847 


286 


9-906018 


155 


9-866829 


441 


10133171 


39 


22 


773018 


286 


905925 


155 


867094 


441 


132906 


38 


23 


773190 


286 


905832 


155 


867358 


441 


} 32642 


37 


24 


773361 


285 


905739 


155 


867623 


441 


132377 


36 


25 


773533 


285 


905645 


155 


867S87 


441 


132113 


35 


26 


773704 


285 


905552 


155 


868152 


440 


131848 


34 


27 


773875 


285 


905459 


155 


868416 


440 


131584 


33 


28 


774046 


285 


905366 


156 


868D80 


440 


131320 


32 


29 


774217 


285 


905272 


156 


868945 


440 


131055 


31 


30 


774388 


284 


905179 


156 


869209 


440 


130791 


30 


31 


9 774558 


284 


9-905085 


156 


9-869473 


440 


10130597 


29 


32 


774729 


284 


904992 


156 


869737 


440 


130263 


28 


33 


774899 


284 


904898 


156 


870001 


440 


129999 


27 


34 


775070 


284 


904804 


156 


870265 


440 


129735 


26 


35 


775240 


284 


904711 


156 


870529 


440 


129471 


25 


36 


775410 


283 


904617 


156 


870793 


440 


129207 


24 


37 


775580 


283 


904523 


156 


871057 


440 


128943 


23 


38 


775750 


283 


904429 


157 


871321 


440 


128679 


22 


39 


775920 


283 


904335 


157 


871585 


440 


128415 


21 


40 


776090 


283 


904241 


157 


871849 


439 


128151 


20 


41 


9776250 


283 


9-904147 


157 


9-872112 


439 


10-127888 


19 


42 


776429 


282 


904053 


157 


872376 


439 


127624 


]H 


43 


776598 


282 


903959 


157 


872640 


439 


127360 


17 


44 


776768 


282 


903864 


157 


872903 


439 


127097 


16 


45 


776937 


282 


903770 


157 


873167 


439 


126833 


15 


46 


777106 


282 


903676 


157 


873430 


439 


126570 


14 


47 


777275 


281 


903581 


157 


873694 


439 


126306 


J 3 


48 


777444 


281 


903487 


157 


873957 


439 


126043 


12 


49 


777613 


281 


903392 


158 


874220 


439 


125780 


11 


50 


777781 


281 


903298 


158 


874484 


439 


125516 


10 


51 


9-777950 


281 


9-903203 


158 


9-874747 


439 


10125253 


9 


52 


778119 


281 


903108 


158 


875010 


439 


124990 


8 


53 


778287 


280 


903014 


158 


875273 


438 


124727 


7 


54 


778455 


280 


9029 J 9 


158 


875536 


438 


124464 


6 


55 


778624 


280 


902824 


158 


875800 


438 


124200 


5 


56 


778792 


280 


902729 


158 


876063 


438 


123937 


4 


57 


778960 


280 


902634 


158 


876326 


438 


123674 


3 


58 


779128 


280 


902539 


159 


876589 


438 


123411 


2 


59 


779295 


279 


902444 


159 


876851 


438 


123149 


1 


60 


779463 


279 


902349 


159 


877114 


438 


122886 






\ Cosine | 



( Sine | 



| Cotang. | 



I Tang. 



53 Degrees. 



ZOGAEITffMIC SINES, COSINES, ETC. (37 Degrees.) 235 



| Sine 


D. 


"9-779463 


279 


779631 


279 


779798 


279 


779966 


279 


780133 


279 


78U300 


278 


780467 


278 


780634 


278 


780801 


278 


780968 


278 


781134 


278 


9781301 


277 


781468 


277 


781634 


277 


781800 


277 


781966 


277 


782132 


277 


782298 


276 


782464 


276 


782630 


276 


782796 


276 


9-782961 


276 


783127 


276 


783292 


275 


783458 


275 


783623 


275 


783788 


275 


783953 


275 


784118 


275 


784282 


274 


784447 


274 


9-784612 


274 


784776 


274 


784941 


274 


785105 


274 


785269 


273 


785433 


273 


785597 


273 


785761 


273 


■ 785925 


273 


786089 


273 


9786252 


272 


786416 


272 


786579 


272 


786742 


272 


786906 


272 


787069 


272 


787232 


271 


787395 


271 


787557 


271 


787720 


271 


9-787883 


271 


788045 


271 


788208 


271 


788370 


270 


7885.32 


270 


788694 


270 


788856 


270 


789018 


270 


789180 


270 


789342 


269 


| Cosine 





Cosine 


1 D. 


Tang. 


D. 


Cotang. 




9902349 


159 


9877114 


438 


10-122886 


60 


902253 


159 


877377 


438 


122623 


59 


902158 


159 


877640 


438 


122360 


58 


902063 


159 


877903 


438 


122097 


57 


901967 


159 


878165 


438 


121835 


56 


901872 


159 


878428 


438 


121572 


55 


90 J 776 


159 


878691 


438 


121309 


54 


90J681 


159 


878953 


437 


121047 


53 


901585 


159 


879216 


437 


120784 


52 


901490 


159 


879478 


437 


120522 


51 


901394 


160 


879741 


437 


120259 


50 


9-901298 


160 


9-880003 


437 


10-119997 


49 


901202 


160 


880265 


437 


119735 


48 


901106 


160 


880528 


437 


119472 


47 


901010 


160 


880790 


437 


119210 


46 


900914 


160 


881052 


437 


118948 


45 


900818 


160 


881314 


437 


118686 


44 


900722 


160 


881573 


437 


118424 


43 


900626 


160 


881839 


437 


118161 


42 


900529 


160 


882101 


437 


117899 


41 


900433 


161 


882363 


436 


117637 


40 


9-900337 


161 


9-882625 


436 


10117375 


39 


900240 


161 


882887 


4:m 


117113 


38 


900144 


161 


883148 


436 


116852 


37 


900047 


161 


883410 


436 


116590 


36 


899951 


161 


883672 


436 


116328 


35 


899854 


161 


883934 


436 


116066 


34 


899757 


161 


884196 


436 


115804 


33 


899660 


161 


884457 


436 


115543 


32 


899564 


161 


884719 


436 


115281 


31 


899467 


162 


884980 


436 


115020 


30 


9-899370 


162 


9-885242 


436 


10114758 


29 


899273 


162 


885503 


436 


114497 


28 


899176 


162 


885765 


436 


114235 


27 


899078 


162 


886026 


436 


113974 


26 


898981 


162 


886288 


436 


113712 


25 


898884 


162 


886549 


435 


113451 


24 


898787 


162 


886810 


435 


113190 


23 


898689 


162 


887072 


435 


112928 


22 


898592 


162 


887333 


435 


112667 


21 


898494 


163 


887594 


435 


112406 


20 


9-898397 


163 


9-887855 


435 


10112145 


19 


898299 


163 


888116 


435 


111884 


18 


898202 


163 


888377 


435 


111623 


17 


898104 


163 


888639 


435 


111361 


16 


898006 


163 


888900 


435 


111100 


15 


897908 


163 


889160 


435 


110840 


14 


897810 


163 


889421 


435 


110579 


13 


897712 


163 


889682 


435 


110318 


12 


897614 


163 


889943 


435 


110057 


11 


897516 


163 


890204 


434 


109796 


10 


9-897418 


164 


9-890465 


434 


10-109535 


9 


897320 


164 


890725 


434 


109275 


8 


897222 


164 


890986 


434 


109014 


7 


897123 


164 


891247 


434 


108753 


6 


897025 


164 


891507 


434 


108493 


5 


89fi926 


164 


891768 


434 


108232 


4 


896828 


164 


892028 


434 


107972 


3 


896729 


164 


892289 


434 


107711 


2 


896631 


164 


892549 


434 


107451 


1 


896532 


164 


892810 


434 


107190 






I 



I Cotang. | 



| Tang. | M. 



52 Degrees. 



236 


(38 Degrees.) 1 


LOGARITj 


UMK! , 


SI2?ES> C 


osmE 


S t ETC. 




M. 


Sine ] 


D. | 


Cosine 


D. \ 


Tang. I 


D. i 


Cotang. 







9-789342 


269 


9896532 


164 


9892810 


434 


10107190 


60 


1 


789504 


269 


896433 


165 


893070 


434 


106930 


59 


o 


789665 


269 


896335 


165 


893331 


434 


106669 


58 


3 


789827 


269 


896236 


165 


893591 


434 


106409 


57 


4 


789988 


269 


896137 


165 


893851 


434 


106149 


56 


5 


790149 


269 


896038 


165 


8941 1 1 


434 


105889 


55 


6 


790310 


268 


895939 


165 


894371 


434 


105629 


54 


7 


790471 


268 


895840 


165 


894632 


433 


105368 


53 


8 


790632 


268 . 


895741 


165 


894892 


433 


105108 


52 


9 


790793 


268 


'893641 


165 


895152 


433 


104848 


51 


10 


790954 


268 


895542 


165 


895412 


433 


104588 


50 


n 


9-791115 


2«8 


9-895443 


166 


9-895672 


433 


10-104328 


49 


12 


791275 


2K7 


895343 


166 


895932 


433 


104068 


48 


13 


791436 


267 


895244 


166 


896192 


433 


103808 


47 


14 


791596 


267 


895145 


166 


896452 


433 


103548 


46 


15 


791757 


267 


895045 


1(56 


896712 


433 


103288 


45 


16 


791917 


267 


894945 


ltj6 


896971 


433 


103029 


44 


17 


792077 


267 


894846 


166 


897231 


433 


102769 


43 


18 


792237 


266 


894746 


166 


897491 


433 


102509 


42 


19 


792397 


266 


894646 


166 


897751 


433 


102249 


41 


20 


792557 


266 


894546 


166 


898010 


433 


101990 


40 


21 


9792716 


266 


9894446 


167 


9-898270 


433 


10-101730 


39 


22 


792876 


266 


894346 


167 


898530 


433 


101470 


38 


23 


793035 


266 


894246 


167 


898789 


433 


101211 


37 


24 


793195 


265 


894146 


167 


899049 


432 


100951 


36 


25 


793354 


265 


894046 


167 


899308 


432 


100692 


35 


26 


793514 


265 


893946 


167 


899568 


432 


100432 


34 


27 


793673 


265 


893846 


167 


899827 


432 


] 00 173 


33 


28 


793832 


265 


893745 


167 


900086 


432 


099914 


32 


29 


793991 


265 


893645 


167 


900346 


432 


099654 


31 


30 


794150 


264 


893544 


167 


900605 


432 


099395 


30 


31 


9-794308 


264 


9-893444 


168 


9-900864 


432 


10099136 


29 


32 


794467 


264 


893343 


168 


901124 


432 


098876 


28 


33 


794626 


264 


893243 


168 


901383 


432 


098617 


27 


34 


794784 


264 


893142 


168 


901642 


432 


098358 


26 


35 


794942 


264 


893041 


168 


901901 


432 


098099 


25 


36 


795101 


264 


892940 


168 


902160 


432 


097840 


24 


37 


795259 


264 


892839 


168 


902419 


432 


097581 


23 


38 


795417 


263 


892739 


168 


902679 


432 


097321 


22 


39 


795575 


263 


892638 


168 


902938 


432 


097062 


21 


40 


795733 


263 


892536 


168 


903197 


431 


096803 


20 


41 


9-795891 


263 


9-892435 


169 


9-903455 


431 


10096545 


19 


42 


796049 


263 


892334 


169 


903714 


431 


096286 


18 


43 


796206 


263 


892233 


169 


903973 


431 


096027 


17 


H 


796364 


262 


892132 


169 


904232 


431 


095768 


16 


45 


796521 


262 


892030 


169 


904491 


431 


095509 


15 


46 


796679 


262 


891929 


169 


904750 


431 


095250 


14 


47 


796836 


262 


891827 


169 


905008 


431 


094992 


13 


48 


796993 


262 


891726 


169 


905267 


431 


094733 


12 


49 


797150 


261 


891624 


169 


905526 


431 


094474 


11 


50 


797307 


261 


891523 


170 


905784 


431 


094216 


10 


51 


9-797464 


261 


9-891421 


170 


9906043 


431 


10093957 


9 


52 


797621 


261 


891319 


170 


906302 


431 


093698 


8 


53 


797777 


261 


891217 


170 


906560 


431 


093440 


7 


54 


797934 


261 


891115 


170 


906819 


431 


093181 


6 


55 


798091 


261 


891013 


170 


907077 


431 


092923 


5 


56 


798247 


261 


890911 


170 


907336 


431 


092664 


4 


57 


798403 


260 


890809 


170 


007594 


431 


092406 


3 


58 


798560 


260 


890707 


170 


907852 


431 


092148 


2 


59 


798716 


260 


890605 


170 


908111 


430 


091889 


1 


60 


798872 
I Cosine 


260 


890503 


170 


908369 


430 


091631 







1 


| Sine 


1 


Cotang. 


1 


U Tanf. 


! * 








6 


L Degre< 


)& 









. LOGARITHMIC SINES, COSINES, ETC. (39 Degrees.) 237 



M. 


| Sine 


! D. 


Cosine 


! D. 


Tang. 


D. 


Cotang. | 







9-798872 


260 


9-890503 


170 


9-908369 


430 


10-091631 


60 


1 


799028 


260 


890400 


171 


908628 


430 


091372 


59 


2 


799184 


260 


890298 


171 


908886 


430 


091114 


58 


3 


799339 


259 


890195 


171 


909144 


430 


090856 


57 


4 


799495 


259 


890093 


171 


909402 


430 


090598 


56 


5 


799651 


259 


889990 


171 


909660 


430 


090340 


55 


6 


799806 


259 


889888 


171 


909918 


430 


090082 


54 


7 


799962 


259 


889785 


171 


910177 


430 


089823 


53 


8 


800J 17 


259 


889682 


171 


910435 


430 


089565 


52 


9 


800272 


258 


889579 


171 


910693 


430 


089307 


51 


10 


800427 


258 


889477 


171 


910951 


430 


089049 


50 


11 


9-800582 


258 


9-889374 


172 


9-911209 


430 


10-088791 


49 


12 


800737 


258 


889271 


172 


911467 


430 


088533 


48 


13 


800892 


258 


889168 


172 


911724 


430 


088276 


47 


14 


801047 


258 


889064 


172 


911982 


430 


088018 


46 


15 


801201 


258 


888961 


172 


912240 


430 


087760 


45 


16 


801356 


257 


888858 


172 


912498 


430 


087502 


44 


17 


801511 


257 


888755 


172 


912756 


430 


087244 


43 


18 


801665 


257 


888651 


172 


913014 


429 


086986 


42 


19 


801819 


257 


888548 


172 


913271 


429 


086729 


41 


20 


801973 


257 


888444 


173 


913529 


429 


086471 


40 


21 


9-802128 


257 


9-888341 


173 


9-913787 


429 


10-086213 


39 


22 


802282 


256 


888237 


173 


914044 


429 


085956 


38 


23 


802436 


256 


888134 


173 


9H302 


429 


085698 


37 


24 


802589 


256 


888030 


173 


914560 


429 


085440 


36 


25 


802743 


256 


887926 


173 


914817 


429 


085 183 


35 


26 


802897 


256 


887822 


173 


915075 


429 


084925 


34 


27 


803050 


256 


887718 


173 


915332 


429 


084668 


33 


28 


803204 


256 


887614 


173 


915590 


429 


084410 


32 


29 


803357 


255 


887510 


173 


915847 


429 


084153 


31 


30 


803511 


255 


887406 


174 


916104 


429 


083896 


30 


31 


9-803664 


255 


9-887302 


174 


9916362 


429 


10-083638 


29 


32 


8038 J 7 


255 


887198 


174 


916619 


429 


083381 


28 


33 


803970 


255 


887093 


174 


916877 


429 


083123 


27 


34 


804123 


255 


886989 


174 


917134 


429 


082866 


26 


35 


804276 


254 


886885 


174 


917391 


429 


082609 


25 


36 


804428 


254 


886780 


174 


917648 


429 . 


082352 


24 


37 


804581 


254 


886676 


174r 


917905 


429 


082095 


23 


38 


804734 


254 


886571 


174 


918163 


428 


081837 


22 


39 


804886 


254 


886466 


174 


918420 


428 


081580 


21 


40 


805039 


254 


886362 


175 


918677 


428 


081323 


20 


41 


9-805191 


254 


9-886257 


175 


9-918934 


428 


10-081066 


19 


42 


805343 


253 


886152 


175 


919191 


428 


0808)9 


18 


43 


805495 


253 


886047 


175 


919448 


428 


080552 


17 


44 


805647 


253 


885942 


175 


919705 


428 


080295 


16 


45 


805799 


253 


885837 


175 


919962 


428 


080038 


15 


46 


805951 


253 


885732 


175 


920219 


428 


079781 


14 


47 


806103 


253 


885627 


175 


920476 


428 


079524 


13 


48 


806254 


253 


885522 


175 


920733 


428 


079267 


12 


49 


806406 


252 


885416 


175 


920990 


428 


079010 


11 


50 


806557 


252 


885311 


176 


921247 


428 


078753 


10 


51 


9-806709 


252 


9-885205 


176 


9921503 


428 


10*078497 


9 


52 


806860 


252 


885100 


176 


921760 


428 


078240 


8 


53 


807011 


252 


884994 


176 


922017 


428 


077983 


7 


54 


807163 


252 


884889 


176 


922274 


428 


077726 


6 


55 


807314 


252 


884783 


176 


922530 


428 


077470 


5 


56 


807465 


251 


884677 


176 


922787 


428 


077213 


4 


57 


807615 


251 


884572 


176 


923044 


428 


076956 


3 


58 


807766 


251 


884466 


176 


923300 


428 


076700 


2 


59 


807917 


251 


884360 


176 


923557 


427 


076443 


1 


60 


808067 


251 


884254 


177 


923813 


427 


076187 






I Cosine 



| Sine 1 



I Cotang. | 



1 Tang. | M. 



50 Degrees. 



288 (40 Degrees.) LOGARITHMIC SINES, COSINES ETC. 



| D. | Cosine \ D. \ Tang. | D. | Cotang. | 






9-808067 


251 


9-884254 


177 


9923813 


427 


10-076187 


60 


1 


808218 


251 


884148 


177 


924070 


427 


075930 


59 


2 


808368 


251 


884042 


177 


924327 


427 


075673 


58 


3 


808519 


250 


883936 


177 


924583 


427 


075417 


57 


4 


808669 


250 


883829 


177 


924840 


427 


075160 


56 


5 


808819 


250 


883723 


177 


925096 


427 


074904 


55 


6 


808969 


250 


883617 


177 


925352 


427 


074648 


54 


7 


809119 


250 


883510 


177 


925609 


427 


074391 


53 


8 


809269 


250 


883404 


177 


925865 


427 


074135 


52 


9 


809419 


249 


883297 


178 


926122 


427 


073878 


51 


]() 


809569 


249 


883191 


178 


926378 


427 


073622 


50 


11 


9809718 


249 


9-883084 


178 


9926634 


427 


10-073366 


49 


12 


809868 


249 


882977 


178 


926890 


427 


073110 


48 


13 


810017 


249 


882871 


178 


927147 


427 


072853 


47 


14 


810167 


249 


882764 


178 


927403 


427 


072597 


46 


15 


810316 


248 


882657 


178 


927659 


427 


072341 


45 


16 


810465 


248 


882550 


178 


927915 


427 


072085 


44 


17 


8 10614 


248 


882443 


178 


928171 


427 


071829 


43 


18 


810763 


248 


882336 


179 


928427 


427 


071573 


42 


19 


810912 


248 


882229 


179 


928683 


427 


071317 


41 


30 


81 J 061 


248 


882121 


179 


928940 


427 


071060 


40 


21 


9-811210 


248 


9-882014 


179 


9-929196 


427 


10*070804 


39 


22 


811358 


247 


881907 


179 


929452 


427 


070548 


38 


23 


811507 


247 


881799 


179 


929708 


427 


070292 


37 


24 


811655 


247 


881692 


179 


929964 


426 


070036 


36 


25 


811804 


247 


881584 


179 


930220 


426 


069780 


35 


26 


811952 


247 


881477 


179 


930475 


426 


069525 


34 


27 


812100 


247 


881369 


179 


930731 


426 


069269 


33 


23 


812248 


247 


881261 


180 


930987 


426 


069013 


32 


29 


812396 


246 


881153 


180 


931243 


426 


068757 


31 


30 


812544 


246 


881046 


180 


931499 


426 


068501 


30 


31 


9812692 


246 


9-880938 


180 


9-931755 


426 


10-068245 


29 


32 


812840 


246 


880830 


180 


932010 


426 


OG7C90 


28 


33 


812988 


246 


880722 


180 


932266 


426 


0C7734 


27 


34 


813135 


246 


880613 


180 


932522 


426 


0C7478 


26 


35 


813283 


246 


880505 


180 


932778 


426 


0G7222 


25 


36 


813430 


245 


880397 


180 


933033 


426 


0CG9G7 


24 


37 


813578 


245 


880289 


181 


933289 


426 


06G711 


23 


38 


813725 


245 


880180 


181 


933545 


426 


06C455 


22 


39 


813872 


245 


880072 


181 


933800 


426 


066200 


21 


40 


814019 


245 


879963 


181 


934056 


426 


065944 


20 


41 


9-814166 


245 


9879855 


181 


9-934311 


426 


10-065689 


19 


42 


814313 


245 


879746 


18/ 


934567 


426 


065433 


18 


43 


814460 


244 


879637 


181 


934823 


426 


065177 


17 


44 


814607 


244 


879529 


181 


935078 


426 


064922 


16 


45 


814753 


244 


879420 


181 


935333 


426 


064667 


15 


46 


814900 


244 


879311 


181 


935589 


426 


064411 


14 


47 


815046 


244 


879202 


182 


935844 


426 


064156 


13 


48 


815193 


244 


879093 


182 


936100 


426 


063900 


12 


49 


815339 


244 


878984 


182 


936355 


426 


063645 


11 


50 


815485 


243 


878875 


182 


936610 


426 


063390 


10 


51 


9-815631 


243 


9-878766 


182 


9936866 


425 


10063134 


9 


52 


815778 


243 


878656 


182 


937121 


425 


062879 


8 


53 


815924 


243 


878547 


182 


937376 


425 


062624 


7 


54 


816009 


243 


878438 


182 


937632 


425 


062368 


6 


55 


816215 


243 


878328 


182 


937887 


425 


062113 


5 


56 


816361 


243 


878219 


183 


938142 


425 


061858 


4 


57 


8! .".507 


242 


878109 


183 


938398 


425 


061602 


3 


58 


816652 


242 


877999 


183 


938653 


425 


061347 


2 


59 


816798 


242 


877890 


183 


938908 


425 


061092 


1 


60 


816943 


242 


877780 


1 183 


939163 


425 


060837 






I Coaine i 



} | Cotang. | 

49 Degrees. 



J Tang. j M, 



LOGARITHMIC SINES, COSINES, ETC. (41 Degrees.) 239 



M. 


| Sine 


1 D. 


Cosine 


D. 


1 Tang. 


1 D. 


Cotang. 







9-816943 


242 


9 877780 


183 


9-939163 


425 


10060837 


60 


1 


817088 


242 


877670 


183 


939418 


425 


060582 


50 


2 


817233 


242 


877560 


183 


939673 


425 


060327 


58 


3 


817379 


242 


877450 


183 


939928 


425 


060072 


57 


4 


817524 


241 


877340 


183 


940183 


425 


059817 


56 


5 


817668 


241 


877230 


184 


940438 


425 


059562 


55 


6 


817813 


241 


877120 


184 


940694 


425 


059306 


54 


7 


817958 


241 


877010 


184 


940949 


425 


059051 


53 


8 


818103 


241 


876899 


184 


941204 


425 


058796 


52 


9 


818847 


241 


876789 


184 


941458 


425 


058542 


51 


10 


818392 


241 


876678 


184 


941714 


425 


058286 


50 


11 


9-818536 


240 


9 876568 


184 


9-941968 


425 


10-058032 


49 


12 


818681 


240 


876457 


184 


942223 


425 


057777 


48 


13 


818825 


240 


876347 


184 


942478 


425 


057522 


47 


14 


818969 


240 


876236 


185 


942733 


425 


057267 


46 


15 


819113 


240 


876125 


185 


942988 


425 


057012 


45 


16 


819257 


240 


876014 


185 


943243 


425 


056757 


44 


17 


819401 


240 


875904 


185 


943498 


425 


056502 


43 


18 


819545 


239 


875793 


185 


943752 


425 


056248 


42 


19 


819689 


239 


875682 


185 


944007 


425 


055993 


41 


20 


819832 


239 


875571 


185 


944262 


425 


055738 


40 


21 


9-819976 


239 


9-875459 


185 


9-944517 


425 


10-055483 


39 


22 


820120 


239 


875348 


185 


944771 


424 


055229 


38 


23 


820263 


239 


875237 


185 


945026 


424 


054974 


37 


24 


820406 


239 


875126 


186 


945281 


424 


054719 


36 


25 


820550 


238 


875014 


186 


945535 


424 


054465 


35 


26 


820693 


238 


874903 


186 


945790 


424 


054210 


34 


27 


820836 


238 


874791 


186 


946045 


424 


053955 


33 


28 


820979 


238 


874680 


186 


946299 


424 


053701 


32 


29 


821122 


238 


874568 


186 


946554 


424 


053446 


31 


30 


821265 


238 


874456 


186 


946808 


424 


053192 


30 


31 


9-821407 


238 


9874344 


186 


9-947063 


424 


10052937 


29 


32 


821550 


238 


874232 


187 


947318 


424 


052682 


28 


33 


821693 


237 


874121 


187 


947572 


424 


052428 


27 


34 


821835 


237 


874009 


187 


947826 


424 


052174 


2e 


35 


821977 


237 


873896 


187 


948081 


424 


051919 


25 


36 


822120 

822262 


237 
237 


873784 
873672 


187 


948336 


424 
424 


051664 
051410 


24 


37 


187 


948590 


23 


38 


822404 


237 


873560 


187 


948844 


424 


051156 


22 


39 


822546 


237 


873448 


187 


949099 


424 


050901 


21 


40 


822688 


236 


873335 


187 


949353 


424 


050647 


20 


41 


9-822830 


236 


9-873223 


187 


9-949607 


424 


10.050393 


19 


42 


822972 


236 


873110 


188 


949862 


424 


050138 


18 


43 


823114 


236 


872998 


188 


950116 


424 


049884 


17 


44 


823255 


236 


872885 


188 


950370 


424 


049630 


16 


45 


823397 


236 


872772 


188 


950625 


424 


049375 


15 


46 


823539 


236 


872659 


188 


950879 


424 


049121 


14 


47 


823680 


235 


872547 


188 


951133 


424 


048867 


13 


48 


823821 


235 


872434 


188 


951388 


424 


048612 


12 


49 


823963 


235 


872321 


188 


951642 


424 


048358 


11 


50 


824104 


235 


872208 


188 


951896 


424 


048104 


10 


51 


9-824245 


235 


9-872095 


189 


9-952150 


424 


10047850 


9 


52 


8243^6 


235 


871981 


189 


952405 


424 


047595 


8 


53 


824527 


235 


871868 


189 


952659 


424 


047341 


7 


54 


824668 


234 


871755 


189 


952913 


424 


047087 


6 


55 


824808 


234 


. 871641 


189 


953167 


423 


046833 


5 


56 


824949 


234 


871528 


189 


953421 


423 


046579 


4 


57 


825090 


234 


871414 


189 


953675 


423 


046325 


3 


58 


825230 


234 


871301 


189 


953929 


423 


046071 


2 


59 


825371 


234 


871187 


189 


954183 


423 


045817 


1 


60 


825511 


234 


871073 


190 


954437 


423 


045563 






| Cosine I 



Sine | | Cotang. | 

48 Degrees. 



| Tang. f M: 



240 (42 Degrees.) LOGARITHMIC SINES, COSINES, ETC. 



M. 


' Sine 


1 D. 


Cosine 


D. 


1 Tang. 


| O. 


| Cotang. 


1 





9-825511 


234 


9-871073 


1 190 


9*954437 


423 


10-045583 


60 


1 


825651 


233 


870960 


1 190 


954691 


423 


045309 


59 


2 


825791 


233 


870846 


190 


954945 


423 


045055 


58 


3 


825931 


233 


870732 


190 


955200 


423 


044800 


57 


4 


826071 


233 


870618 


190 


955454 


423 


044546 


56 


5 


826211 


233 


870504 


190 


955707 


423 


044293 


55 


6 


826351 


233 


870390 


190 


955961 


423 


044039 


54 


7 


826491 


233 


870276 


190 


956215 


423 


043785 


53 


8 


826631 


233 


870161 


190 


956469 


423 


043531 


52 


9 


826770 


232 


870047 


191 


956723 


423 


043277 


51 


10 


826910 


232 


869933 


191 


956977 


423 


043023 


50 


11 


9-827049 


232 


9-869818 


191 


9-957231 


423 


10-042769 


49 


12 


827189 


232 


869704 


191 


957485 


423 


042515 


48 


13 


827328 


232 


869589 


191 


957739 


423 


042261 


47 


14 


827467 


232 


869474 


191 


957993 


423 


042007 


46 


15 


827606 


232 


869360 


191 


958246 


423 


041754 


45 


16 


827745 


232 


869245 


191 


958500 


423 


041500 


44 


17 


827884 


231 


869130 


191 


958754 


423 


041246 


43 


18 


828023 


231 


869015 


192 


959008 


423 


040992 


42 


19 


828162 


231 


868900 


192 


959262 


423 


040738 


41 


20 


828301 


231 


868785 ■ 


192 


959516 


423 


040484 


40 


21 


9828439 


231 


9868670 


192 


9-959769 


423 


10040231 


39 


22 


828578 


231 


868555 


192 


960023 


423 


039977 


38 


23 


828716 


231 


868440 


192 


960277 


423 


039723 


37 


24 


828855 


230 


868324 


192 


960531 


423 


039469 


36 


25 


828993 


230 


868209 


192 


960784 


423 


039216 


35 


26 


829131 


230 


868093 


192 


961038 


423 


038962 


34 


27 


829269 


230 


867978 


193 


961291 


423 


038709 


33 


28 


829407 


230 


867862 


193 


961545 


423 


038455 


32 


29 


829545 


230 


867747 


193 


961799 


423 


038201 


31 


30 


829683 


230 


867631 


193 


962052 


423 


037948 


30 


31 


9829821 


229 


9-867515 


193 


9-962306 


423 


10037694 


29 


32 


829959 


229 


867399 


193 


962560 


423 


037440 


28 


33 


830097 


229 


867283 


193 


962813 


423 


037187 


27 


34 


830234 


229 


867167 


193 


963067 


423 


036933 


26 


35 


830372 


229 


867051 


193 


963320 


423 


036680 


25 


36 


830509 


229 


866935 


194 


963574 


423 


036426 


24 


37 


830646 


229 


866819 


194 


963827 


423 


036173 


23 


38 


830784 


229 


866703 


194 


964081 


423 


035919 


2S 


39 


830921 


228 


866586 


194 


964335 


423 


035665 


21 


40 


831058 


228 


866470 


194 


964588 


422 


035412 


20 


41 


9-831195 


228 


9-866353 


194 


9-964842 


422 


10035158 


19 


42 


831332 


228 


866237 


194 


965095 


422 


034905 


18 


43 


831469 


228 


866120 


194 


965349 


422 


034651 


17 


44 


831606 


228 


866004 


195 


965602 


422 


034398 


16 


45 


831742 


228 


865887 


195 


965855 


422 


034145 


15 


46 


831879 


228 


865770 


195 


966109 


422 


033891 


14 


47 


832015 


227 


865653 


195 


966362 


422 


033638 


13 


48 


832152 


227 


865536 


195 


966616 


422 


033384 


12 


49 


832288 


227 


865419 


195 


966869 


422 


033131 


11 


50 


832425 


227 


865302 


195 


967123 


422 


032877 


10 


51 


9832561 


227 


9'865185 


195 


9-967376 


422 


10032624 


9 


52 


832697 


227 


865068 


195 


967629 


422 


032371 


8 


53 


832833 


227 


864950 


195 


967883 


422 


032117 


7 


54 


832969 


226 


864833 


196 


968136 


422 


031864 


6 


55 


833105 


226 


864716 


196 


968389 


422 


031611 


5 


56 


833241 


226 


864598 


196 


968643 


422 


031357 


4 


57 


833377 


226 


864481 


196 


968896 


422 


031104 


3 


58 


833512 


226 


864363 


196 


969149 


422 


030851 


2 


59 


833648 


226 


864245 


196 


969403 


422 


030597 


1 


60 


833783 


226 


864127 


196 


969656 


422 


030344 






l Cosine » 



» Sine | 



j Cotang. \ 



Tang. i tl 



47 Degrees. 



LOGARITHMIC SINES* CO SIXES, ETC. (43 Degrees.) 241 



M 


| Sine 


1 D. 


| Cosine 


1 D. 


Tang. 


1 D. 


Cotang-. 







9-833783 


226 


9-864127 


196 


9-969656 


422 


10-030344 


1 60 


1 


833919 


225 


864010 


196 


969909 


422 


030091 


59 


2 


834054 


225 


863892 


197 


970162 


422 


029838 


58 


3 


834189 


225 


863774 


197 


970416 


422 


029584 


57 


4 


834325 


225 


863656 


197 


970669 


422 


029331 


56 


5 


834460 


225 


863538 


197 


970922 


422 


029078 


55 


o 


834595 


225 


863419 


197 


971175 


422 


028825 


54 


7 


834730 


225 


863301 


197 


971429 


422 


028571 


53 


8 


834865 


225 


863183 


197 


971682 


422 


028318 


52 


9 


834999 


224 


863064 


197 


971935 


422 


028065 


51 


10 


835134 


224 


862946 


198 


972188 


422 


027812 


50 


11 


9835269 


224 


9-862827 


198 


9-972441 


422 


10-027559 


49 


12 


835403 


224 


862709 


198 


972694 


422 


027306 


48 


13 


835538 


224 


862590 


198 


972948 


422 


027052 


47 


14 


835672 


224 


862471 


198 


973201 


422 


026799 


46 


15 


835807 


224 


862353 


198 


973454 


422 


026546 


45 


1G 


835941 


224 


862234 


198 


973707 


422 


026293 


44 


1? 


836075 


223 


862115 


198 


973960 


422 


026040 


43 


16 


836209 


223 


861996 


198 


974213 


422 


025787 


42 


19 


836343 


223 


861877 


198 


974466 


422 


025534 


41 


20 


836477 


223 


861758 


199 


974719 


422 


025281 


40 


21 


9-836611 


223 


9-861638 


199 


9-974973 


422 


10-025027 


39 


22 


836745 


223 


861519 


199 


975226 


422 


024774 


38 


23 


836878 


223 


861400 


199 


975479 


422 


024521 


37 


24 


837012 


222 


861280 


199 


975732 


422 


024268 


36 


25 


837146 


222 


861161 


199 


975985 


422 


024015 


35 


26 


837279 


222 


861041 


199 


976238 


422 


023762 


34 


27 


837412 


222 


860922 


199 


976491 


422 


023509 


33 


28 


837546 


222 


860802 


199 


976744 


422 


023256 


32 


29 


837679 


222 


860682 


200 


976997 


422 


023003 


31 


30 


837812 


222 


860562 


200 


977250 


422 


022750 


30 


31 


9-837945 


222 


9-860442 


200 


9-977503 


422 


10-022497 


29 


32 


838078 


221 


860322 


200 


977756 


422 


022244 


28 


33 


838211 


221 


860202 


200 


978009 


422 


021991 


27 


34 


838344 


221 


860082 


200 


978262 


422 


021738 


26 


35 


838477 


221 


859962 


200 


978515 


422 


021485 


25 


3G 


838610 


221 


859842 


200 


978768 


422 


021232 


24 


37 


838742 


221 


859721 


201 


979021 


422 


020979 


23 


as 


838875 


221 


859601 


201 


979274 


422 


020726 


22 


39 


839007 


221 


859480 


201 


979527 


422 


020473 


21 


40 


839140 


220 


859360 


201 . 


979780 


422 


020220 


20 


41 


9-839272 


220 


9-859239 


201 


9980033 


422 


10019967 


19 


42 


839404 


220 


859119 


201 


980286 


422 


019714 


18 


43 


839536 


220 


858998 


201 


980538 


422 


019462 


17 


44 


839668 


220 


858877 


201 


980791 


421 


019209 


16 


45 


839800 


220 


858756 


202 


981044 


421 


018956 


15 


46 


839932 


220 


858635 


202 


981297 


421 


018703 


14 


47 


840064 


219 


858514 


202 


981550 


421 


018450 


13 


48 


840196 


219 


858393 


202 


981803 


421 


018197 


1.2 


49 


840328 


219 


858272 


202 


982056 


421 


017944 


11 


50 


840459 


219 


858151 


202 


982309 


421 


017691 


10 


51 


9-840591 


219 


9-858029 


202 


9-982562 


421 


10-017438 


9 


52 


840722 


219 


857908 


202 


982814 


421 


017186 


8 


53 


840854 


219 


857786 


202 


983067 


421 


016933 


7 


54 


840985 


219 


857665 


203 


983320 


421 


016680 


6 


55 


841116 


218 


857543 


203 


983573 


421 


016427 


5 


56 


841247 


218 


857422 


203 


983826 


421 


016174 


4 


57 


841378 


218 


857300 


203 


984079 


421 


015921 


3 


58 


841509 


218 


857178 


203 


984331 


421 


015669 


2 


59 


841640 


218 


857056 


203 


984584 


421 


015416 


1 


60 


841771 


218 


856934 ' 


203 


984837 


421 


015163 






i Comae | 



| Sine j 



| Cotang. | 



I Tang. | M. 



46 Degrees. 



242 (44 Degrees.) LOGARITHMIC SINES, CO SIXES, ETC. 



| Sine | 






9-841771 


218 


1 


841902 


218 


2 


842033 


218 


3 


842163 


217 


4 


842294 


217 


5 


842424 


217 


6 


842555 


217 


7 


842685 


217 


8 


842815 


217 


9 


842946 


217 


10 


843076 


217 


11 


9-843206 


216 


12 


843336 


216 


13 


843466 


216 


14 


843595 


216 


15 


843725 


216 


1G 


843855 


216 


17 


843984 


216 


18 


844114 


215 


19 


844243 


215 


20 


844372 


215 


21 


9-844502 


215 


22 


844631 


215 


23 


844760 


215 


24 


844889 


215 


25 


845018 


215 


2G 


845147 


215 


27 


845276 


214 


28 


845405 


214 


29 


845533 


214 


30 


845662 


214 


31 


9845790 


214 


32 


845919 


214 


33 


846047 


214 


34 


846175 


214 


35 


846304 


214 


36 


846432 


213 


37 


846560 


213 


38 


846688 


213 


39 


846816 


213 


40 


846944 


213 


41 


9-847071 


213 


12 


847199 


213 


43 


847327 


213 


44 


847454 


212 


45 


847582 


212 


46 


847709 


212 


47 


847836 


212 


48 


847964 


212 


49 


848091 


212 


50 


848218 


212 


51 


9-848345 


212 


52 


848472 


211 


53 


848599 


211 


54 


848726 


211 


55 


848852 


211 


56 


848979 


211 


57 


849106 


211 


58 


849232 


211 


59 


849359 


211 


60 


849485 


211 



I Cosine | D> | Tang. | D. | Cotang. | 



| Cosine | 



9-856934 
856812 
856690 
856568 
856446 
856323 
856201 
856078 
855956 
855833 
855711 

9-855588 
855465 
855342 
855219 
855096 
854973 
854850 
854727 
854603 
854480 

9-854356 
854233 
854109 
853986 
853862 
853738 
853614 
853490 
853366 
853242 

9853118 
852994 
852869 
852745 
852620 
852496 
852371 
852247 
852122 
851997 

9-851872 
851747 
851622 
851497 
851372 
851246 
851121 
850996 
850870 
850745 

9-850619 
850493 
850368 
850242 
850116 
849990 
849864 
849738 
849611 
849485 

| Sins 



203 
203 
204 
204 
204 
204 
204 
204 
204 
204 
205 
205 
205 
205 
205 
205 
205 
205 
206 
206 
206 
206 
206 
206 
206 
206 
206 
207 
207 
207 
207 
207 
207 
207 
207 
207 
208 
208 
208 
208 
208. 
208 
208 
208 
209 
209 
209 
209 
209 
209 
209 
209 
210 
210 
210 
210 
210 
210 
210 
210 
210 



9-984837 
985090 
985343 
985596 



986101 
986354 
986607 



987112 

987365 

9-987618 

987871 
988123 
988376 

988629 



989134 



989640 
989893 

9-990145 
990398 
990651 
990903 
991156 
991409 
991662 
991914 
992167 
992420 

9*992672 
992925 
993178 
993430 
993683 
993936 
994189 
994441 
994694 
994947 

9-995199 
995452 
995705 
995957 
996210 
996463 
996715 
996968 
997221 
997473 

9-997726 
997979 
998231 
998484 
998737 
998989 
999242 
999495 
999748 
10000000 



421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 
421 



10-015163 
014910 
014G57 
014404 
014152 
013899 
013046 
013393 
013140 
012888 
012635 

10*012382 
012129 
011877 
11624 
011371 
011118 
010866 
010613 
010360 
010107 

10-009855 
009602 
009349 
009097 
008844 
008591 
008338 
008086 
007833 
007580 

10-007328 
007075 
006822 
006570 
006317 
006064 
005811 
005559 
005306 
005053 

10-004801 
004548 
004295 
004043 
003790 
003537 
003285 
003032 
002779 
002527 

10002274 
002021 
001769 
001516 
001263 
001011 
000758 
000505 
000253 
000000 



I 



| Cotang. | 



I Tang. | M. 



45 Degrees, 



TABLE XIV. 
NATURAL SIGNS AND TANGENTS. 



244 



NATURAL SINES. 



0° j 


1° 


2° ! 


3° 


4° 


5° 1 


6° 


7° 


•000 0000-017 4524 


•034 8995-052 3360 


•069 7565 


•087 1557 


•104 5285 


121 8693 


2909 j 7432 


•035 1902 


6264 


•070 0467 


4455 


8178 


122 1581 


5818 -018 0341 


4809 


9169 


3368 


7353 


•105 1070 


4468 


8727 


3249 


7716 


•053 2074 


6270 


•088 0251 


3963 


7355 


•001 1636 


6158 


•0360623 


4979 


9171 


3148 


6856 


123 0241 


4544 


9066 3530 


7883 


•071 2073 


6046 


9748 


3128 


7453 


•019 1974 


6437 


•054 0788 


4974 


8943 


•106 2641 


6015 


•002 0362 


4883 


9344 


3693 


7876 


•089 1840 


5533 


8901 


3271 


7791 


037 2251 


6597 


•0720777 


4738 


8425 


1241788 


6180 


•020 0699 


5158 


9502 


3678 


7635 


•107 1318 


4674 


9089 


3608 


8065 


•055 2406 


6580 


090 0532 


4210 


7560 


•0031998 


6516 


•0380971 


5311 


9481 


3429 


7102 


125 0446 


4907 


9424 


3878 


8215 


•073 2382 


6326 


9994 


3332 


7815 


•021 2332 


6785 


•056 1119 


5283 


9223 


•108 2885 


6218 


•004 0724 


5241 


9692 


4024 


8184 


•091 2119 


5777 


9104 


3633 


8149 


•039 2598 


6928 


•074 1085 


5016 


8669 


126 1990 


6542 


•0221057 


5505 


9832 


3986 


7913 


•1G9 1560 


4875 


9451 


3965 


8411 


•057 2736 


6887 


•0920809 


4452 


7761 


•0052360 


6873 


•040 1318 


5640 


9787 


3706 


7343 


127 0646 


5268 


9781 


4224 


8544 


•075 2688 


6602 


•1100234 


3531 


8177 


•0232690 


7131 


•058 1448 


5589 


9499 


3126 


6416 


•OOtf 1086 


5598 


•041 0037 


4352 


8489 


•093 2395 


6017 


9302 


3995 


8506 


2944 


7256 


•0761390 


5291 


8908 


•128 2186 


6904 


•0241414 


5850 


•0590160 


4290 


8187 


•111 1799 


5071 


9813 


4322 


8757 


3064 


7190 


•094 1083 


4689 


7956 


•007 2721 


7230 


•042 1663 


5967 


•077 0091 


3979 


7580 


•129 0841 


5630 


•025 0138 


4569 


8871 


2991 


6675 


•112 0471 


3725 


8589 


3046 


7475 


•060 1775 


5891 


9771 


3361 


6609 


•008 1448 


5954 


•043 0382 


4678 


8791 


•095 2666 


6252 


9494 


4357 8862 


3288 


7582 


•078 1691 


5562 


9142 


•130 2378 


7265 


•0ft6 1769 


6194 


•061 0485 


4591 


8458 


•113 2032 


5262 


•0090174 


4677 


9100 


3389 


7491 


•096 1353 


4922 


8146 


3083 


7685 


•044 2006 


6292 


•079 0391 


4248 


7812 


•131 1030 


5992 


•027 0493 


4912 


9196 


3290 


7144 


•1140702 


3913 


8900 


3401 


7818 


•062 2099 


6190 


•097 0039 


3592 


6797 


•010 1809 


6309!'C<5 0724 


5002 


9090 


2934 


6482 


9681 


4718 


9216! 3630 


7905 


•080 1989 


5829 


9372 


•132 2564 


7627 


•028 2124J 6536 


•0630808 


4889 


8724 


•115 2261 


5447 


•Oil 0535 


5032. 9442 


3711 


7788 


•098 1619 


5151 


8330 


3444 


7940/046 2347 


6614 


•081 0687 


4514 


8040 


1331213 


6353 


•0290847! 5253 


9517 


3587 


7408 


•116 0929 


4096 


9261 


3755: . 8159 


•064 2420 


6486 


•099 0303 


3818 


6979 


012 2170 


6662-047 1065 


5323 


9385 


3197 


6707 


9862 


5079 


9570 j 3970 


8226 


•082 2284 


6092 


9596 


134 2744 


7987 


•030 2478! 6876 


•065 1129 


5183 


8986 


•117 2485 


5627 


•013 0896 


53851 9781 


4031 


8082 


•100 1881 


5374 


8509 


3805 


8293 1 -048 2687 


6934 


•C«3 0981 


4775 


8263 


•135 1392 


6713 


•031 1200! 5592 


9836 


3880 


7669 


•118 1151 


4274 


9622 


4108 8498 


•066 2739 


6778 


•101 0563 


4040 


7156 


•014 2530 


7015*049 1403 


5641 


9677 


3457 


6928 


•1360038 


5439 


9922 1 4308 


8544 


•084 2576 


6351 


9816 


2919 


8348 


•0322830': 7214 


•067 1446 


5474 


9245 


119 2704 


5801 


•015 1256 


5737 -050 0119 


4349 


8373 


•102 2138 


5593 


8683 


4165 


8644 3024 


7251 


•085 1271 


5032 


8481 


•137 1564 


7073 -033 1552 5929 


•068 0153 


4169 


7925 


•120 1368 


4445 


9982! 4459 8835 


3055 


7067 


•103 0819 


4256 


7327 


•016 28901 7366-0511740 


5957 


9966 


3712 


| 7144 


•138 0208 


5799,-034 0274 4645 


8859 


•086 2864 


6605 


12* 0031 


3089 


8707! 3181 7550 


•069 1761 


5762 


9499 


2919 


5970 


•017 1616 6088 -052 0455 


4663 


8660 


•104 2392 


5806 


8850 


4524 8995 3360 


7565 


•087 1557 


5285 


8693 


•13? 1731 


89° 88° i 87° 


86° 


85° 


84° 


83° 


82° 








NAT. < 


30SINE. 









NATURAL SINES. 



245 



8° 


9° 


10° 


11° 


12° 


13° 


14° 


15° 


139 1731 


156 4345 


173 6482 


•190 8090 


•207 9117 


224 9511 


•241 9219 


258 8190 


4612 


7218 


9346 


•191 0945 


•208 1962 


225 2345 


242 2041 


•259 1000 


7492 


•157 0091 


174 2211 


3801 


4807 


5179 


4863 


3810 


140 0372 


2963 


5075 


6656 


7652 


8013 


7685 


6619 


3252 


5836 


7939 


9510 


•209 0497 


226 0846 


243 0507 


9428 


6132 


870S 


175 0803 


•192 2365 


3341 


3680 


3329 


260 2237 


9012 


•158 1581 


3667 


5220 


6186 


6513 


6150 


5045 


141 1892 


4453 


6531 


8074 


9030 


9346 


8971 


7853 


4772 


7325 


9395 


•193 0928 


•210 1874 


227 2179 


2441792 


261 0662 


7651 


•159 0197 


176 2258 


3782 


4718 


5012 


4613 


3469 


1420531 


3069 


5121 


6636 


7561 


7844 


7433 


6277 


3410 


5940 


7984 


9490 


•211 0405 


228 0677 


245 0254 


9085 


6289 


8812 


177 0847 


•1942344 


3248 


3509 


3074 


262 1892 


9168 


•160 1683 


3710 


5197 


6091 


6341 


5894 


469C- 


143 2047 


4555 


6573 


8050 


8934 


9172 


8713 


750C 


4926 


7426 


9435 


•195 0903 


•212 1777 


229 2004 


246 1533 


•263 0312 


7805 


•161 0297 


178 2298 


3756 


4619 


4835 


4352 


3118 


1440684 


3167 


5160 


6609 


7462 


7666 


7171 


5925 


3562 


6038 


8022 


9461 


•213 0304 


230 0497 


9990 


8730 


6440 


8909 


179 0884 


•196 2314 


3146 


3328 


•247 2809 


•264 1536 


9319 


•162 1779 


3746 


5166 


5988 


6159 


5627 


4342 


145 2197 


4650 


6607 


8018 


8829 


8989 


8445 


7147 


5075 


7520 


9469 


•197 0870 


•2141671 


•231 1819 


•248 1263 


9952 


7953 


•163 0390 


•180 2330 


3722 


4512 


4649 


4081 


•265 2757 


146 0830 


3260 


5191 


6573 


7353 


7479 


6899 


5561 


3708 


6129 


8052 


9425 


•215 0194 


•2320309 


9716 


8366 


6585 


8999 


•181 0913 


•198 2276 


3035 


3138 


•249 2533 


•266 1170 


9463 


•1641868 


3774 


5127 


5876 


5967 


5350 


3973 


147 2340 


4738 


6635 


7978 


8716 


8796 


8167 


6777 


5217 


7607 


9495 


•199 0829 


•2161556 


•233 1625 


•250 0984 


9581 


8094 


•165 0476 


182 2355 


3679 


4396 


4454 


3800 


•267 2384 


148 0971 


3345 


5215 


6530 


7236 


7282 


6616 


5187 


3848 


6214 


8075 


9380 


•217 0076 


•234 0110 


9432 


7989 


6724 


9082 


•183 0935 


•200 2230 


2915 


2938 


•251 2248 


•268 0792 


9601 


1661951 


3795 


5080 


5754 


5766 


5063 


3594 


149 2477 


4819 


6654 


7930 


8593 


8594 


7879 


6396 


5353 


7687 


9514 


•201 0779 


•218 1432 


•235 1421 


•252 0694 


9198 


8230 


•167 0556 


•184 2373 


3629 


4271 


4248 


3508 


•269 2000 


150 1106 


3423 


5232 


6478 


7110 


7075 


6323 


4801 


3981 


6291 


8091 


9327 


9948 


9902 


9137 


7602 


6857 


9159 


•185 0949 


•202 2176 


•219 2786 


•236 2729 


•2531952 


•270 0403 


9733 


•168 2026 


3808 


5024 


5624 


5555 


4766 


3204 


151 2608 


4894 


6666 


7873 


8462 


8381 


7579 


6004 


5484 


7761 


9524 


•203 0721 


•2201300 


•237 1207 


•2540393 


8805 


8359 


•169 0628 


•186 2382 


3569 


4137 


4033 


3206 


•271 1605 


1521234 


3495 


5240 


6418 


6974 


6859 


6019 


4404 


4109 


6362 


8098 


9265 


9811 


9684 


8832 


7204 


6984 


9228 


•187 0956 


•204 2113 


•221 2648 


•238 2510 


•255 1645 


•2720003 


9858 


•170 2095 


3813 


4961 


5485 


5335 


4458 


2802 


•153 2733 


4961 


6670 


7808 


8321 


8159 


7270 


5601 


5607 


7828 


9528 


•205 0655 


•222 1158 


•239 0984 


•256 0082 


8400 


8482 


171 0694 


•188 2385 


3502 


3994 


3808 


2894 


•2731198 


•154 1356 


3560 


5241 


6349 


6830 


6633 


5705 


3997 


4230 


6425 


8098 


9195 


9666 


9457 


8517 


6794 


7101 


9291 


•189 0954 


•206 2042 


•223 2501 


•240 2280 


•257 1328 


9592 


997$ 


172 2156 


3811 


4888 


5337 


5104 


4139 


•2742390 


•155 2851 


5022 


6667 


7734 


8172 


7927 


695C 


5187 


mi 


7887 


9522 


■207 0580 


•224 1007 


•241 0751 


976C 


7984 


859$ 


•173 0752 


190 237£ 


3426 


3842 


3574 


•258 2570 


•275 0781 


•156 147S 


3617 


5234 


t 6272 


6676 


6396 


5381 


3577 


434£ 


► 6482 


809C 


> 9117 


9511 


921S 


819C 


6374 


81° 


80° 


79° 78° 


77° 


76° 


75° 


74° 








NAT. 


COSINE 









246 



NATURAL SINES. 



16° 


17° 


18° 


19° 


20° 


21° 


22° | 


23° | 


/ 


.275 C374 


292 3717 


309 0170 


325 5682 


342 0201 


•658 3679 


374 6066-390 73111 60 


9170 


6499 


2936 


8432 


2935 


6395 


8763 9989 59 


276 1965 


9280 


5702 


326 1182 


5668 


9110 


3751459.3912666 58 


47G1 


293 2061 


8468 


3932 


8400 


359 1825 


4156 5343; 57 


7556 


4842 


310 1234 


6681 


343 1133 


4540 


6852 8019' 56 


277 0352 


7623 


, 3999 


9430 


3865 


7254 


9547 -392 0695 ! 55 


3147 


294 0403 


6764 


327 2179 


6597 


9968 


376 2243 3371 


54 


5941 


3183 9529 


4928 


9329 


•360 2682 


4938 1 6047 


53 


8736 


5963 


•311 2294 


7676 


•3442060 


5395 


7632 1 8722 


52 


•278 1530 


8743 


5058 


328 0424 


4791 


8108 


377 0327 


393 1397 


51 


4324 


295 1522 


C1 7822 


3172 


7521 


•361 0821 


3021 


4071 


50 


7118 


4302 


•312 0586 


5919 


•345 0252 


3534 


5714 


6745 


49 


9911 


7081 


3349 


8666 


2982 


6246 


8408 


9419 


48 


•279 2704 


9859 


6112 


329 1413 


5712 


8958 


•378 1101 


394 2093 


47 


5497 


296 2638 


8875 


4160 


8441 


•362 1669 


3794 


4766 


46 


8290 


5416 


•313 1638 


6906 


•3461171 


4380 


6486 


7439 


45 


•280 1083 


8194 


4400 


9653 


3900 


7091 


9178 


•395 0111. 


44 


3875 


•297 0971 


7163 


•330 2398 


6628 


9802 


•379 1870 


2783 


43 


6667 


3749 


9925 


5144 


9357 


•363 2512 


4562 


5455 


42 


9459 


6526 


•314 2686 


7889 


•347 2085 


5222 


7253 


8127 


41 


•281 2251 


9303 


5448 


•331 0634 


4812 


7932 


9944 


•396 0798 


40 


5042 


•298 2079 


8209 


337y 


7540 


•3640641 


380 2634 


3468 


39 


7833 


4856 


•315 0969 


6123 


•348 0267 


3351 


5324 


6139 


38 


•282 0624 


7632 


3730 


8867 


2994 


6059 


8014 


8809 


37 


3415 


•299 0408 


6490 


•332 1611 


5720 


8768 


•381 0704 


•397 1479 


36 


6205 


3184 


9250 


4355 


8447 


•365 1476 


3393 


4148 


35 


8995 


5959 


•316 2010 


7098 


•349 1173 


4184 


6082 


6818 


34 


•2831785 


8734 


4770 


9841 


3898 


6891 


8770 


9486 


33 


4575 


•300 1509 


7529 


•3332584 


6624 


9599 


•3821459 


•398 2155 


32 


7364 


4284 


•317 0288 


5326 


9349 


•366 2306 


4147 


4823 


31 


•2840153 


7058 


3047 


8069 


•350 2074 


5012 


6834 


7491 


30 


2942 


9832 


5805 


•3340810 


4798 


7719 


9522 


.399 0158 


29 


5731 


•301 2606 


8563 


3552 


7523 


•367 0425 


•383 2209 


2825 


28 


8520 


5380 


•318 1321 


6293 


•351 0246 


3130 


4895 


5492 


27 


•285 1308 


8153 


4079 


9034 


2970 


5836 


7582 


8158 


26 


4096 


•3020926 


6836 


•3351775 


5693 


8541 


•384 0268 


•400 0825 


25 


6884 


3699 


9593 


4516 


8416 


•368 1246 


2953 


3490 


24 


9671 


6471 


•319 2350 


7256 


•3521139 


3950 


5639 


6156 


23 


•2862458 


9244 


5106 


9996 


3862 


6654 


8324 


8821 


22 


5246 


•303 2016 


7863 


•336 2735 


6584 


9358 


•385 1008 


•401 1486 


21 


8032 


4788 


•320 0619 


5475 


9306 


•369 2061 


3693 


4150 


20 


•287 0819 


7559 


3374 


8214 


•3532027 


4765 6377 


6814 


19 


3605 


•3040331 


6130 


•337 0953 


4748 


7468 


9060 


9478 


18 


6391 


3102 


8885 


3691 


7469 


•370 0170 


•386 1744 


•402 2141 


17 


9177 


5872 


•321 1640 


6429 


•3540190 


2872 


4427 


4804 


16 


•288 1963 


8643 


4395 


9167 


2910 


5574 


7110 


7467 


15 


4748 


•305 1413 


7149 


•338 1905 


5630 


8276 


9792 


•403 0129 


14 


7533 


4183 


9903 


4642 


8350 


•371 0977 


•387 2474 


2791 


13 


•289 0318 


6953 


•3222657 


7379 


•355 1070 


3678 


5156 


5453 


12 


3103 


9723 


5411 


•339 one 


3789 


6379 


7837 


8114 


11 


5887 


•306 2492 


8164 


2852 


6508 


9079 


•388 0518 


•404 0775 


10 


8671 


5261 


•3230917 


5586 


922C 


•3721780 


3199 


3436 


9 


•290 1455 


803C 


3670 


8325 


•3561944 


4479 


5880 


6096 


8 


4239 


•307 0798 


6422 


•340 106C 


4662 


7179 


8560 


8756 


7 


7022 


356C 


9174 


3796 


738C 


9878 


•389 1240 


•405 1416 


6 


9805 


6334 


•3241926 


6531 


•357 0097 


•373 2577 


3919 


4075 


5 


•291 2588 


9102 


4678 


9265 


2814 


5275 


6598 


6734 


4 


5371 


' 308 186S 


7429 


•341 200C 


5531 


7972 


9277 


9392 


3 


815£ 


463e 


> -325 0180 


4734 


8248 


•374 0671 


•390 1955 


•406 2051 


2 


•292093? 


740J 


1 2931 


7468 


•358 0964 336S 


4633 


470S 


1 


3717 


•309 017( 


) 5682 


•342 0201 


367S 


606C 


1 7311 


736C 





73° 72° b 71° \ 70° 


69° 


68° 


! 67° 


66° 


/ 








NAT. 


COSINE. 











NATURAL SIXES. 



247 



/ 


24° 





•406 7366 


1 


•407 0024 


2 


2681 


3 


5337 


4 


7993 


5 


•408 0649 


6 


3305 


7 


5960 


8 


8615 


9 


•409 1269 


10 


3923 • 


11 


6577 


12 


9230 


13 


•410 1883 


14 


4536 


15 


7189 


16 


9811 


17 


•4112492 


IS 


5144 


19 


7795 


20 


•412 0445 


21 


3096 


22 


5745 


23 


8395 


24 


•413 1044 


25 


3693 


26 


6342 


27 


8990 


28 


•414 1638 


29 


4285 


30 


6932 


31 


9579 


32 


•415 2226 


33 


4872 


31 


7517 


35 


•416 0163 


36 


9^08 


37 


6453 


33 


8097 


39 


•417 0741 


40 


3385 


41 


6028 


42 


8671 


43 


•418 1313 


44 


3956 


45 


6597 


46 


9233 


47 


•419 1880 


48 


4521 


49 


7161 


50 


9801 


51 


•420 2441 


52 


5080 


53 


7719 


54 


•421 0358 


55 


2996 


56 


5634 


57 


8272 


53 


•4220909 


59 


3546 


60 


6183 


/ 


65° 



25° ' 

■422 6183 
8819 

423 1455 
4090 
6725 
9360 

424 1994 
4628 
7262 
9895 

425 2528 
5161 
7793 

426 0425 
3056 
5637 
8318 

427 0949 
3579 
6208 
8838 

428 1467 
4095 
6723 
9351 

429 1979 
46061 
7233 
9859 

430 2485 
5111 
7736 

431 0361 
2986 
5610 
8234 

•432 0857 
3481 
6103 
8726 

433 1348 
3970 
6591 
9212 

■434 1832 
4453 
7072 
9692 

•435 2311 
4930 
7548 

■436 0166 
2784 
5401 
8018 

■437 0634 
3251 
5866 
8482 

•438 1097 
3711 

64° 



26° 


27° 


28° 1 


•438 3711 


•453 9905 


•469 4716 • 


6326 


•454 2497 


7284 • 


8940 


5088 


9S52 


•439 1553 


7679 


•470 2419 


4166 


•455 0269 


4986 


6779 


2859 


7553- 


9392 


5449 


•4710119 


•440 2004 


8038 


2685 


4615 


•456 0627 


5250 


7227 


3216 


7815- 


9838 


5804 


•472 0380 


•441 2448 


8392 


2944 


5059 


•457 0979 


5508 


7668 


3566 


8071- 


•442 0278 


6153 


•473 0634 


2887 


8739 


3197 


5496 


•458 1325 


5759 


8104 


3910 


8321- 


•443 0712 


6496 


•474 0882 


3319 


9080 


3443 


5927 


•459 1665 


6004 


8534 


4248 


8564 


•444 1140 


6832 


•475 1124 


3746 


9415 


3683 


6352 


•460 1998 


6242 


8957 


4580 


8801 


•445 1562 


7162 


•476 1359 


4167 


9744 


3917 


6771 


•461 2325 


6474 


9375 


4906 


9031 


•446 1978 


7486 


•477 1588 


4581 


•462 0066 


4144 


7184 


2646 


6700 


9786 


5225 


9255 


•447 2388 


7804 


•478 1810 


4990 


•463 0382 


4364 


7591 


2960 


6919 


•448 0192 


5538 


9472 


2792 


8115 


•479 2026 


5392 


•464 0692 


4579 


7992 


3269 


7131 


•449 0591 


5845 


9683 


3190 


8420 


•480 2235 


5789 


•465 0996 


4786 


8387 


3571 


7337 


•450 0984 


6145 


9888 


3582 


8719 


•481 2438 


6179 


•466 1293 


4987 


8775 


3866 


7537 


•451 1372 


6439 


•482 0086 


3967 


9012 


2634 


6563 


•467 1584 


5182 


9158 


4156 


7730 


•4521753 


6727 


•483 0277 


4347 


9298 


2824 


6941 


•4681869 


5370 


9535 


4439 


7916 


•453 2128 


7009 


•484 0462 


4721 


9578 


3007 


7313 


•469 2147 


5552 


9905 


4716 


8096, 


63° 


62° 


61° J 



29° ! 

484 8096 

485 06401 
3184| 
5727! 
8270! 

486 0812 
3354 
5895 
8436 

487 0977 
3517 
6057 
8597 

488 1136 
3674 
6212 
8750 

489 1288 
3825 
6361 
8897 

490 1433 
3968 
6503 
9038 

491 1572 
4105 
6638 
9171 

492 1704 
4236 
6767 
9298 

■493 1829 
4359 
6889 
9419 

■494 1948 
4476 
7005 
9532| 

•495 2060 
4587 
7113 
9639 

•496 2165 
4690 
7215 
9740 

•497 2264 
4787 
7310 
9833 

•498 2355 
4877 
7399 
9920 

•499 2441 
4961 
7481 

•500 0000 

60° 



30° 

500 0000 
2519 
5037 
7556 

501 0073 
2591 
510: 
7624 

■502 0140 
2655 
5170 
7685 

•503 0199 
2713 
522' 
7740 

•504 0252 
2765 
5276 
7788 

505 0298 
2809 
5319 
7828 

506 0338 
2846 
5355 
7863 

507 0370 
2877 
5384 
7890 

508 0396 
2901 
5406 
7910 

■509 0414 
2918 
5421 
7924 

•510 0426 
2928 
5429 



31< 

515 0381 
2874 
5367 
7859 

516 0351 
2842 
5333 
7824 

517 0314 
2804 
5293 
7782 

518 0270 
2758 
5246 
7733 

519 0219 
2705 
5191 
7676 

520 0161 
2646 
5130 
7613 

521 0096 
2579 
5061 
7543 

•522 0024 
2505 
4986 
7466 
9945 

•523 2424 
4903 
7381 
9859 

•524 2336 
4813 
7290 
9766 

•525 2241 
471 



7930 


7191 


511 0431 


9665 


2931 


•526 2139 


5431 


4613 


7930 


7085 


512 0429 


9558 


2927 


•527 2030 


5425 


4502 


7923 


6973 


513 0420 


9443 


2916 


•528 1914 


5413 


4383 


7908 


6853 


514 0404 


9322 


2899 


•529 1790 


5393 


4258 


7887 


6726 


515 0381 


9193 


59° 


58° 



NAT. COSINB. 



248 



NATURAL SINES* 



32° 


33° 


34° 


35° 


36° 


37° 


38° 


39° 


/ 


529 9193 


544 6390 


559 1929 


573 5764 


587 7853 


601 8150 


•615 6615 


•629 3204 


60 


530 1659 


8830 


4340 


8147 


588 0206 


602.0473 


8907 


5464 


59 


4125 


545 1269 


6751 


574 0529 


2558 


2795 


•6161198 


7724 


58 


6591 


3707 


9162 


2911 


4910 


5117 


3489 


9983 


57 


9057 


6145 


560 1572 


5292 


7262 


7439 


5780 


•630 2242 


56 


531 1521 


8583 


3981 


7672 


9613 


9760 


8069 


4500 


55 


3986 


546 1020 


6390 


575 0053 


589 1964 


603 2080 


•617 0359 


6758 


54 


6450 


3456 


8798 


2432 


4314 


4400 


2648 


9015 


53 


8913 


5892 


561 1206 


4811 


6663 


6719 


4936 


•631 1272 


52 


532 1376 


8328 


3614 


7190 


9012 


9038 


7224 


3528 


51 


3839 


547 0763 


6021 


9568 


•590 1361 


.604 1356 


9511 


5784 


50 


6301 


3198 


8428 


576.1946 


3709 


3674 


•618 1798 


8039 


49 


8763 


5632 


562 0834 


4323 


6057 


5991 


4084 


•632 0293 


43 


533 1224 


8066 


3239 


6700 


8404 


8308 


6370 


2547 


47 


3685 


•548 0499 


5645 


9076 


•591 0750 


•605 0624 


8655 


4800 


46 


6145 


2932 


8049 


•577 1452 


3096 


2940 


•619 0939 


7053 


45 


8605 


5365 


•563 0453 


3827 


5442 


5255 


3224 


9306 


44 


534 1065 


7797 


2857 


6202 


7787 


7570 


5507 


•633 1557 


43 


3523 


•549 0228 


5260 


8576 


•592 0132 


9884 


7790 


3809 


42 


5982 


2659 


7663 


•578 0950 


2476 


•606 2198 


•620 0073 


6059 


41 


8440 


5090 


•564 0066 


3323 


4819 


4511 


2355 


8310 


40 


535 0898 


7520 


2467 


5696 


7163 


6824 


4636 


•634 0559 


39 


3355 


9950 


4869 


8069 


9505 


9136 


6917 


2808 


38 


5812 


•550 2379 


7270 


•579 0440 


•593 1847 


•607 1447 


9198 


5057 


37 


8268 


4807 


9670 


2812 


4189 


3758 


•621 1478 


7305 


36 


536 0724 


7236 


•565 2070 


5183 


6530 


6069 


3757 


9553 


35 


3179 


9663 


4469 


7553 


8871 


8379 


6036 


•6351800 


34 


5R34 


•551 2091 


6868 


9923 


•5941211 


•608 0689 


8314 


4046 


33 


8089 


4518 


9267 


•580 2292 


3550 


2998 


•6220592 


• 6292 


32 


537 0543 


6944 


■566.1665 


4661 


5889 


5306 


2870 


8537 


31 


2996 


9370 


4062 


7030 


8228 


7614 


5146 


•6360782 


30 


5449 


•552 1795 


6459 


9397 


•595 0566 


9922 


7423 


3026 


29 


7902 


4220 


8856 


•581 1765 


2904 


•609 2229 


9698 


5270 


28 


•538 0354 


6645 


567 1252 


4132 


5241 


4535 


•623 1974 


7513 


27 


2806 


9069 


3648 


6498 


7577 


6841 


4248 


9756 


26 


5257 


•553 1492 


6043 


8864 


9913 


9147 


6522 


•637 1998 


25 


7708 


3915 


8437 


•582 1230 


•596 2249 


•610 1452 


8796 


4240 


24 


•5390158 


- 6338 


•568 0832 


3595 


4584 


3756 


•624 1069 


6481 


23 


2608 


8760 


3225 


5959 


691S 


6060 


3342 


8721 


22 


5058 


•5541182 


5619 


8323 


9252 


8363 


5614 


•638 0961 


21 


7507 


3603 


8011 


•5830687 


•597 1586 


•611 0666 


7885 


3201 


20 


9955 


6024 


•569 0403 


3050 


3919 


2969 


•625 015C 


5440 


19 


•540 2103 


8444 


2795 


5412 


6251 


5270 


2427 


7678 


18 


4851 


•555 0864 


5187 


7774 


8583 


7572 


4696 


9916 


17 


7298 


3283 


7577 


•584 0136 


•598 0915 


. 9873 


6966 


•639 2153 


16 


9745 


5702 


9968 


2497 


3246 


•612 2173 


9235 


4390 


15 


•541 2191 


8121 


•570 2357 


4857 


5577 


4473 


•626 1503 


6626 


14 


4637 


•556 0539 


4747 


7217 


7906 


6772 


8771 


8862 


13 


7082 


2956 


7136 


9577 


•599 0236 


9071 


6038 


•640 1097 


12 


9527 


5373 


9524 


•585 1986 


2565 


•613 1369 


8305 


3332 


11 


•542 1971 


7790 


•571 1912 


4294 


4893 


366f 


•627 0571 


5566 


10 


4415 


•557 0206 


4299 


6652 


7221 


5964 


2837 


7799 


9 


6859 


2621 


6686 


9010 


9549 


8260 


5102 


•641 0032 


S 


9302 


5036 


9073 


•586 1367 


•600 1876 


•614 0556 


7366 


2264 


7 


•543 1744 


7451 


•572 1459 


3724 


4202 


2852 


9631 


4496 


6 


4187 


9865 


3844 


6080 


6528 


5147 


•628 1894 


6728 


5 


6628 


•558 2279 


6229 


8435 


8854 


7442 


4157 


8958 


4 


9069 


4692 


8614 


•587 0790 


•601 1179 


9736 


6420 


•6421189 


3 


•5441510 


7105 


•5730998 


3145 


3503 


•615 2029 


8682 


3418 


2 


3951 


9517 


3381 


5499 


5827 


4322 


•629 0943 


5647 


1 


6390 


•559 1929 


5764 


7853 


8150 


6615 


8204 


7876 





57° 


56° 


65° 


54° 

NAT. ( 


53° 

JOSINP 


52° 


51° 


50° 


/ 



NATURAL SINES. 



249 



40° 


41° 


42° 


43° 


44° 1 


45° 


46° 


47° 


/ 


642 7876 


6560590 


669 1306 


681 9984 


694 6584| 


707 1068 


719 3398 


731 3537 


60 


643 0104 


2785 


3468 


682 2111 


8676 


3124 


6418 


5521 


59 


2332 


4980 


5628 


4237 


695 0767 


5180 


7438 


7503 


58 


4559 


7174 


7789 


6363 


2858 


7236 


9457 


9486 


57 


6785 


9367 


9948 


8489 


4949 


9291 


7201476 


7321467 


56 


9011 


657 1560 


670 2108 


683 0613 


7039 


708 1345 


3494 


3449 


55 


6441236 


3752 


4266 


2738 


9128 


3398 


5511 


5429 


54 


3461 


5944 


6424 


4861 


6961217 


5451 


7528 


7409 


53 


5685 


8135 


8582 


6984 


2305 


7504 


9544 


9388 


52 


7909 


658 0326 


671 0739 


9107 


5392 


9556 


721 1559 


733 1367 


51 


645 0132 


2516 


2895 


684 1229 


7479 


709 1607 


3574 


3345 


50 


2355 


4706 


5051 


3350 


9565 


3657 


5589 


5322 


49 


4577 


6895 


7206 


5471 


697 1651 


5707 


7602 


7299 


48 


6798 


9083 


9361 


7591 


3736 


7757 


9615 


9275 


47 


9019 


059 1271 


672 1515 


9711 


5821 


9806 


•72213281 


734 1250 


46 


6461240 


3458 


3668 


685 1830 


7905 


•710 1854 


364$ 


3225 


45 


3460 


5645 


5821 


3948 


9988 


3901 


5651 


5199 


44 


5679 


7831 


7973 


6066 


•698 2071 


5948 


7661 


7173 


43 


7898 


660 0017 


673 0125 


8184 


4153 


7995 


9671 


9146 


42 


647 0116 


2202 


2276 


686 0300 


6234 


•711 0041 


•723 1681 


•7351118 


41 


2334 


4386 


4427 


2416 


8315 


2086 


3690 


3090 


40 


4551 


6570 


6577 


4532 


•699 0396 


4130 


5698 


5061 


39 


6767 


8754 


8727 


6647 


2476 


6174 


7705 


7032 


38 


8984 


•661 0936 


•674 0876 


8761 


4555 


8218 


9712 


9002 


37 


648 1199 


3119 


3024 


■687 0875 


6633 


•712 0260 


•7241719 


•736 0971 


36 


3414 


5300 


5172 


2988 


8711 


2303 


3724 


2940 


35 


5628 


7482 


7319 


5101 


•700 0789 


4344 


5729 


4908 


34 


7842 


9662 


9466 


7213 


2866 


6385 


7734 


6875 


33 


649 0056 


•662 1842 


•675 1612 


9325 


4942 


8426 


9738 


8842 


32 


2268 


4022 


3757 


•688 1435 


7018 


•713 0465 


•725 1741 


•737 0808 


31 


4480 


6200 


5902 


3546 


9093 


2504 


3744 


2773 


30 


6692 


837*9 


8046 


5655 


•701 1167 


4543 


5746 


4738 


29 


8903 


•663 0C57 


•676 0190 


7765 


3241 


6581 


7747 


6703 


28 


•650 1114 


2734 


2333 


9873 


5314 


8618 


9748 


8666 


27 


3324 


4910 


4476 


•689 1981 


7387 


•714 0655 


•7261748 


•738 0629 


26 


5533 


7087 


6618 


4089 


9459 


2691 


3748 


2592 


25 


7742 


9262 


8760 


6195 


•702 1531 


4727 


5747 


4553 


24 


9951 


•6641437 


•677 0901 


8302 


3601 


6762 


7745 
9743 


6515 


23 


•6512158 


3612 


3041 


•690 0407 


5672 


8796 


8475 


22 


4366 


5785 


5181 


2512 


7741 


•715 0830 


•727 1740 


•739 0435 


21 


6572 


7959 


7320 


4617 


9811 


2863 


3736 


2394 


20 


8778 


•665 0131 


9459 


6721 


•703 1879 


4895 


5732 


4353 


19 


•6520984 


2304 


•678 1597 


8824 


3947 


6927 


7728 


6311 


18 


3189 


4475 


3734 


•691 0927 


6014 


8959 


9722 


8268 


17 


5394 


6646 


5871 


3029 


8081 


•7160989 


•728 1716 


•740 0225 


16 


7598 


8817 


8007 


5131 


•704 0147 


3019 


3710 


2181 


15 


9801 


•666 0987 


•679 0143 


7232 


2213 


5049 


5703 


4137 


14 


•653 2004 


3156 


2278 


9332 


4278 


7078 


7695 


6092 


13 


4206 


5325 


4413 


•692 1432 


6342 


9106 


9686 


8046 


12 


6408 


7493 


6547 


3531 


8406 


•717-1134 


•7291677 


•741 000C 


11 


8609 


9661 


8681 


5630 


•705 0469 


3161 


3668 


1958 


10 


•654 0810 


•667 1828 


•680 0813 


7728 


2532 


5187 


5657 


390£ 


9 


3010 


3994 


2946 


9825 


4594 


7218 


7646 


5857 


8 


5209 


61 6C 


5078 


•693 1922 


665E 


9238 


9635 


7808 


7 


7408 


8326 


7209 


4018 


8716 


► ■7181262 


•730 1623 


975$ 


6 


9607 


•668 049C 


9339 


6114 


•706 0776 


> 3287 


361C 


•742170$ 


5 


•655 1804 


265£ 


•681 1469 


820£ 


283£ 


> 531C 


5597 


365$ 


5 4 


4002 


4818 


3599 


•694 0304 


489, 


I 733c 


758S 


5606 


5 3 


6198 


6981 


5728 


239$ 


695c 


5 935£ 


> 9568 


755- 


I 2 


839£ 


9144 


t 7856 


4491 


9011 


L -719 137' 


•731 1555 


; 9501 


I 1 


•656 059C 


•669 1306 


) 99841 6584 


\ -707 106f 


J 339$ 


\ 3531 


•7431441 


I 


49° 


48° 


47° 1 46° 


45° 


44° 


43° 


42° 


/ 








NAT. ( 


:osinb 











250 



NATURAL SINES- 



48° 


49° 


•7431448 


•7547096 


3394 


9004 


5340 


•755 0911 


7285 


2818 


9229 


4724 


•744 1173 


6630 


3115 


8535 


5058 


•756 0439 


6999 


2342 


8941 


4246 


•745 0881 


6148 


2821 


8050 


4760 


9951 


6699 


•757 1851 


8636 


3751 


•746 0574 


5650 


2510 


7548 


4446 


9446 


6382 


•758 1343 


8317 


3240 


•747 0251 


5136 


2184 


7031 


4117 


8926 


6049 


•759 0820 


7981 


2713 


9912 


4606 


•748 1842 


6498 


3772 


8389 


5701 


•7600280 


7629 


2170 


9557 


4060 


•749 1484 


5949 


3411 


7837 


5337 


9724 


7262 


•761 1611 


9187 


3497 


•7501111 


5383 


3034 


. 7268 


4957 


9152 


6879 


•762 1036 


8800 


2919 


•751 0721 


4802 


2641 


6683 


4561 


8564 


6480 


•763 0445 


8398 


2325 


•752 0316 


4204 


2233 


6082 


4149 


7960 


6065 


9838 


7980 


•7641714 


9894 


3590 


•7531808 


5465 


3721 


7340 


5634 


9214 


7546 


•765 1087 


9457 


2960 


•754 1368 


4832 


3278 


6704 


5187 


8574 


7096 


•766 0444 


41° 


40° 



50° \ 

766 0444 ! 
2314: 

4183) 
6051 I 
7918, 
9785 i 
•767 1652 j 
3517: 
5382 1 
7246 

9110 | 

•768 0973 
2835 
4697 
6558 
8418 

■7690278 
2137 
3996 
5853 
7710 
9567 

•770 1423 
3278 
5132 
6986 
8840 

■771 0692 
2544 
4395 
6246 
8096 
9945 

•7721794 
3642 
5489 
7336 
9182 

•773 1027 
2872 
4716 
6559 
8402 

•7740244 
2086 
3926 
5767 
7606 
9445 

•775 1283 
3121 
4957 
6794 
8629 

•7760464 
2298 
4132 
5965 
7797 
9629 

•777 1460 

39° 



51° | 


52° 


53° 


54° 


•777 1460 


•788 0108 


•798 6355 


•809 0170 


3290 


1898 


8105 


1879 


5120 


3688 


9855 


3588 


6949 


5477 


•799 1604 


5296 


8777 


7266 


3352 


7004 


•778 0604 


9054 


5100 


8710 


2431 


•789 0841 


6847 


•810 0416 


4258 


2627 


8593 


2122 


6084 


4413 


•800 0338 


3826 


7909 


6198 


2083 


5530 


9733 


7983 


3827 


7234 


•779 1557 


9767 


5571 


8936 


3380 


•7901550 


7314 


•811 0638 


5202 


3333 


9056 


2339 


7024 


5115 


•801 0797 


4040 


8845 


6896 


2538 


5740 


•780 0665 


8676 


4278 


7439 


2485 


•791 0456 


6018 


9137 


4304 


2235 


7756 


•812 083*5 


6123 


4014 


9495 


2532 


7940 


5792 


•802 1232 


4229 


9757 


7569 


2969 


5925 


•781 1574 


9345 


4705 


7620 


3390 


•7921121 


6440 


9314 


5205 


2896 


8175 


•813 1008 


7019 


4671 


9909 


2701 


8833 


6445 


•8031642 


4393 


•7820646 


8218 


3375 


6084 


2459 


9990 


5107 


7775 


4270 


•793 1762 


6838 


9466 


6082 


3533 


• 8569 


•8141155 


7892 


5304 


•8040299 


2844 


9702 


7074 


2028 


4532 


•7831511 


8843 


3756 


6220 


3320 


•794 0611 


5484 


7906 


5127 


2379 


7211 


9593 


6935 


4146 


8938 


•815 1278 


8741 


5913 


•805 0664 


2963 


•7840547 


7678 


2389 


4647 


2352 


9444 


4113 


6330 


4157 


•795 1208 


5837 


8013 


5961 


2972 


7560 


9695 


7764 


4735 


9283 


•8161376 


9566 


6497 


•8061005 


3056 


•785 1368 


8259 


2726 


4736 


3169 


•796 0020 


4446 


6416 


4970 


1780 


■ 6166 


8094 


6770 


3540 


7885 


9772 


8569 


5299 


9603 


•817 1449 


•7860367 


7058 


•807 1321 


3125 


2165 


8815 


3038 


4801 


3963 


•797 0572 


4754 


6476 


5759 


2329 


6470 


8151 


7555 


4084 


8185 


9824 


9350 


5839 


9899 


•818 1497 


•787 1145 


7594 


•808 1612 


3169 


2939 


9347 


3325 


4841 


4732 


•798 1100 


5037 


6512 


6524 


2853 


6749 


8182 


8316 


4604 


8460 


9852 


•788 0108 


6355 


•809 0170 


•819 1520 


38° 


| 37° 


36° 


35° 



NAT. COSINE. 



NATURAL SINES. 



251 



55° 


56° 


57° 


58° 


59° 


60° 


-819 1520 


•829 0376 


•838 6706 


•848 0481 


•857 1673 


•866 0254 


3189 


2002 


8290 


2022 


3171 


1708 


4856 


3628 


9873 


3562 


4668 


3161 


6523 


5252 


•839 1455 


5102 


6164 


4614 


8189 


6877 


3037 


6641 


7660 


6066 


9854 


8500 


4618 


8179 


9155 


7517 


•820 1519 


•830 0123 


6199 


9717 


•858 0649 


8967 


3183 


1745 


7778 


•849 1254 


2143 


•867 0417 


4846 


3366 


9357 


2790 


3635 


1866 


6509 


4987 


•840 0936 


4325 


5127 


3314 


8170 


6607 


2513 


5860 


6619 


4762 


9832 


8226 


4090 


7394 


8109 


6209 


•821 1492 


9845 


5666 


8927 


9599 


7655 


3152 


•831 1463 


7241 


•850 0459 


•859 1088 


9100 


4811 


3080 


8816 


1991 


2576 


•868 0544 


6469 


4696 


•8410390 


3522 


4064 


1988 


8127 


6312 


1963 


5053 


5551 


3431 


9784 


7927 


3536 


6582 


7037 


4874 


8221440 


9541 


5108 


8111 


8523 


6315 


3096 


•8321155 


6679 


9639 


•860 0007 


7756 


4751 


2768 


8249 


•851 1167 


1491 


9196 


6405 


4380 


9819 


2693 


2975 


•869 0636 


8659 


5991 


•842 1388 


• 4219 


4457 


2074 


9712 


7602 


2956 


5745 


5939 


3512 


8231364 


9212 


4524 


7269 


7420 


4949 


3015 


•833 0822 


6091 


8793 


8901 


6386 


4666 


2430 


7657 


•852 0316 


•861 0380 


7821 


6316 


4038 


9222 


1839 


1859 


9256 


7965 


5646 


•843 0787 


3360 


3337 


•8700691 


9614 


7252 


2351 


4881 


4815 


2124 


•824 1262 


8858 


3914 


6402 


6292 


3557 


2909 


•834 0463 


5477 


7921 


- 7768 


4989 


4556 


2068 


7039 


9440 


9243 


6420 


6202 


3672 


8600 


■853 0958 


•862 0717 


7851 


7847 


5275 


•8440161 


2476 


2191 


9281 


9491 


6877 


1720 


3992 


3664 


•871 0710 


•825 1135 


8479 


3279 


5508 


5137 


2138 


2778 


•835 0080 


4838 


T023 


6608 


3566 


4420 


1680 


6395 


8538 


8079 


4993 


6062 


3279 


7952 


•8540051 


9549 


6419 


7703 


4878 


9508 


1564 


•863 1019 


7844 


9343 


6476 


•845 1064 


3077 


2488 


9269 


•826 0983 


8074 


2618 
£L72 


4588 


3956 


•872 0693 


2622 


9670 


6099 


5423 


2116 


4260 


•836 1266 


5726 


7609 


6889 


3538 


5897 


2862 


7278 


9119 


8355 


4960 


7534 


4456 


8830 


•855 0627 


9820 


6381 


9170 


6050 


•846 0381 


2135 


•8641284 


7801 


•827 0806 


7643 


1932 


3643 


2748 


9221 


2440 


9236 


3481 


5149 


4211 


•873 0640 


4074 


•837 0827 


5030 


6655 


5673 


2058 


5708 


2418 


6579 


8160 


7134 


3475 


7340 


4009 


8126 


9664 


8595 


4891 


8972 


5598 


9673 


•856 1168 


•865 0055 


6307 


•828 0603 


7187 


•847 1219 


2671 


1514 


7722 


2234 


8775 


2765 


4173 


2973 


9137 


3864 


•838 0363 


4309 


5674 


4430 


•874 0550 


5493 


1950 


5853 


7175 


5887 


1963 


7121 


3536 


7397 


8675 


7344 


3375 


8749 


5121 


8939 


•857 0174 


8799 


4786 


•829 0376 


6706 


•848 0481 


1673 


•866 0254 


6197 


34° 


33° 


32° 


31° 


30° 


29° 



61° 

874 6197 
7607 
9016 

875 0425 
1832 
3239 
4645 
6051 
7455 
8859 

■876 0263 
1665 
3067 
4468 
5868 
7268 
8666 

■877 0064 
1462 
2858 
4254 
5649 
7043 
8437 
9830 

•878 1222 
2613 
4004 
5394 
6783 
8171 
9559 

•879 0946 
2332 
3717 
5102 
6486 
7869 
9251 

•880 0633 
2014 
3394 
4774 
6152 
7530 
8907 

•881 0284 
1660 
3035 
4409 
5782 
7155 
8527 
9898 

•882 1269 
2638 
4007 
5376 
6743 
8110 
9476 
28° 



NAT. COSINE. 



252 



NATURAL SINES. 



62° 


63° 


04° 


65° 


66° 


67° 


68° 


•882 9476 


.891 0065 


•898 7940 


•906 3078 


•913 5455 


•920 5049 


•927 1839 


•8830841 


1385 


9215 


4307 


6637 


6185 


2928 


2206 


2705 


•899 0489 


5535 


7819 


7320 


4016 


3569 


4024 


1763 


6762 


9001 


8455 


5104 


4933 


5342 


3035 


7989 


•9140181 


9589 


6191 


6295 


6659 


4307 


9215 


1361 


.921 0722 


7277 


7656 


7975 


5578 


•907 0440 


2540 


1854 


8363 


9017 


9291 


6848 


1665 


3718 


2986 


9447 


•8840377 


•892 0606 


8117 


2888 


4895 


4116 


•928 0531 


1736 


1920 


9386 


4111 


6072 


5246 


1614 


3095 


3234 


•900 0654 


5333 


7247 


6375 


2696 


4453 


4546 


1921 


6554 


8422 


7504 


3778 


5810 


5858 


3188 


7775 


9597 


8632 


4858 


7166 


7169 


4453 


8995 


•915 0770 


9758 


5938 


8522 


8480 


5718 


•908 0214 


1943 


•922 0884 


7017 


9876 


9789 


6982 


1432 


3115 


2010 


8096 


•885 1230 


•8931098 


8246 


2649 


4286 


3134 


9173 


2584 


2406 


9508 


3866 


5456 


4258 


•9290250 


3936 


3714 


•901 0770 


5082 


6626 


5381 


1326 


5288 


5021 


2031 


6297 


7795 


6503 


2401 


6639 


6326 


3292 


7511 


8963 


7624 


3475 


7989 


7632 


4551 


8725 


•916 0130 


8745 


4549 


9339 


8936 


5810 


9938 


1297* 


9865 


5622 


•886 0688 


•8940240 


7068 


•909 1150 


2462 


•923 0984 


6694 


2036 


1542 


8325 


2361 


3627 


2102 


77.65 


3383 


2844 


9582 


3572 


4791 


3220 


8835 


4730 


4146 


•9020838 


4781 


5955 


4336 


9905 


6075 


5446 


2092 


5990 


7118 


5452 


•9300974 


7420 


6746 


3347 


7199 


8279 


6567 


2042 


8765 


8045 


4600 


8406 


9440 


7682 


3109 


•887 0108 


9344 


5853 


9613 


•917 0601 


8795 


4176 


1451 


•895 0641 


7105 


•910 0819 


1760 


9908 


5241 


2793 


1938 


8356 


2024 


2919 


•9241020 


6306 


4134 


3234 


9606 


3228 


4077 


2131 


7370 


5475 


4529 


•903 0856 


4432 


6234 


3242 


8434 


6815 


5824 


2105 


5635 


6391 


4351 


9496 


8154 


7118 


3353 


6837 


7546 


5460 


•931 0558 


9492 


8411 


4600 


8038 


8701 


6568 


1619 


•888 0830 


9703 


5847 


9238 


9855 


7676 


2679 


2166 


.896 0994 


7093 


•911 0438 


•918 1009 


8782 


3739 


3503 


2285 


8338 


1637 


2161 


9888 


4797 


4838 


3575 


9582 


2835 


3313 


•925 0993 


5855 


6172 


4864 


•9040825 


4033 


4464 


2097 


6912 


7506 


6153 


2068 


5229 


5614 


3201 


7969 


8839 


7440 


3310 


6425 


6763 


4303 


9024 


•889 0171 


8727 


4551 


7620 


7912 


5405 


•9320079 


1503 


•897 0014 


5792 


8815 


9060 


6506 


1133 


2834 


1299 


7032 


•912 0008 


•919 0207 


7606 


2186 


4164 


2584 


8271 


1201 


1353 


8706 


3238 


5493 


3868 


9509 


2393 


2499 


9805 


4290 


6822 


5151 


•905 0746 


3584 


3644 


•926 0902 


5340 


8149 


6433 


1983 


4775 


4788 


2000 


6390 


9476 


7715 


3219 


5965 


5931 


3096 


7439 


•890 0803 


8996 


4454 


7154 


7073 


4192 


8488 


2128 


•898 0276 


5688 


8342 


8215 


5286 


9535 


3453 


1555 


6922 


9529 


9356 


6380 


•933 0582 


4777 


2834 


8154 


•913 0716 


•920 0496 


7474 


1628 


6100 


4112 


9386 


1902 


1635 


8566 


2673 


7423 


5389 


•906 0618 


3087 


2774 


9658 


3718 


8744 


6665 


1848 


4271 


3912 


•927 0748 


4761 


•891 0065 


7940 


3078 


5455 


5049 


1839 


5804 


27° 


26° 


25° 


24° 


23° 


22° 


21° 1 



I' 



NAT. COSINE. 



NATURAL SINES. 



253 



69° 

•933 5804 
6846 
7888 
8928 
9968 

•9341007 
2045 
3082 
4119 
5154 
6189 
7223 
8257 
9289 

•9350321 
1352 
2382 
3412 
4440 
5468 
6495 
7521 
8547 
9571 

-936 0595 
1618 
2641 
3662 
4683 
5703 
6722 
7740 
8758 
9774 

•937 0790 
1806 
2820 
3833 
4846 
5858 
6869 
7880 



•938 0906 ! 
1913 ■ 
2920 i 
3925 = 
4930 
5934 



70° 


71° 


•939 6926 


•945 5186 


7921 


6132 


8914 


7078 


9907 


8023 


•940 0899 


8968 


1891 


9911 


2881 


•946 0854 


3871 


1795 


4860 


2736 


5848 


3677 


6835 


4616 


7822 


5555 


8808 


6493 


9793 


7430 


941 0777 


8366 


1760 


9301 


2743 


•947 0236 


3724 


1170 


4705 


2103 


5686 


3035 


6665 


3966 


7644 


4897 


8621 


5827 


9598 


6756 


•942 0575 


7684 


1550 


8612 


2525 


9538 


3498 


•948 0464 


4471 


1389 


5444 


2313 


6415 


3237 


7386 


4159 


8355 


5081 


9324 


6002 


•943 0293 


6922 


1260 


7842 


2227 


8760 


3192 


9678 


4157 


•949 0595 


5122 


1511 


6085 


2426 


7048 


3341 


8010 


4255 


8971 


5168 


9931 


6080 


•944 0890 


6991 


1849 


7902 


2807 


8812 


3764 


9721 


4720 


•950 0629 



6938 


5675 


1536 


7940 


6630 


2443 


8942 


7584 


3348 


9943 


8537 


4253 


939 0943 


9489 


5157 


1942 


•945 0441 


6061 


2940 


1391 


6963 


3938 


2341 


7865 


4935 


3290 


8766 


5931 


4238 


9666 


6926 


5186 


•951 0565 


20° 


19° 


18° 



72° 

•951 0565 
1464 
2361 
3258 
4154 
5050 
5944 
6838 
7731 
8623 
9514 

952 0404 
1294 
2183 
3071 
3958 
4844 
5730 
6615 
7499 
8382 
9264 

953 0146 
1027 
1907 
2786 
3664 
4542 
5418 
6294 
7170 
8044 
8917 
9790 

■954 0662 
1533 
2403 
"3273 
4141 
5009 
5876 
6743 
7608 
8473 
9336 

•955 0199 
1062 
1923 
2784 
3646 
4502 
5361 
6218 
7074 
7930 
8785 
9639 

•9560492 
1345 
2197 
3048 

17° 



73° 1 


74° 


75° 


•956 3048 


•961 2617 


•965 9258 


3898 


3418 


•966 0011 


4747 


4219 


0762 


5595 


5019 


1513 


6443 


5818 


2263 


7290 


6616 


3012 


8136 


7413 


3761 


8981 


8210 


4508 


9825 


9005 


5255 


957 0669 


9800 


6001 


1512 


•962 0594 


6746 


2354 


1387 


7490 


3195 


2180 


8234 


4035 


2972 


8977 


4875 


3762 


9718 


5714 


4552 


•967 0459 


6552 


5342 


1200 


7389 


6130 


1939 


8225 


6917 


2678 


9060 


7704 


3415 


9895 


8490 


4152 


•958 0729 


9275 


4888 


1562 


•963 0060 


5624 


2394 


0843 


6358 


3226 


1626 


7092 


4056 


2408 


7825 


4886 


3189 


8557 


5715 


3969 


9288 


6543 


4748 


•968 0018 


7371 


5527 


0748 


8197 


6305 


1476 


9023 


7081 


2204 


9848 


7858 


2931 


•959 0672 


8633 


3658 


1496 


9407 


4383 


2318 


•964 0181 


5108 


3140 


0954 


5832 


3961 


1726 


6555 


4781 


2497 


7277 


5600 


3268 


7998 


6418 


4037 


.8719 


7236 


4806 


9438 


8053 


5574 


•9690157 


8869 


6341 


0875 


9684 


7108 


1593 


•960 0499 


7873 


2309 


1312 


8638 


3025 


2125 


9402 


3740 


2937 


•965 0165 


4453 


3748 


0927 


5167 


4558 


1689 


5879 


5368 


2449 


6591 


6177 


3209 


7301 


6984 


3968 


8011 


7792 


4726 


8720 


8598 


5484 


9428 


9403 


6240 


•970 0135 


•961 0208 


6996 


0842 


1012 


7751 


1548 


1815 


8505 


2253 


2617 


9258 


2957 


16° 


15° 


14° 



NAT. COSINE. 



254 



NATURAL SINES. 



76° 1 


77° 


•970 2957 


•974 3 7 01 


3660 


4355 


4363 


5008 


5065 


5660 


5766 


6311 


6466 


6962 


7165 


7612 


7863 


8261 


8561 


8909 


9258 


9556 


9953 


•975 0203 


•971 0649 


0849 


1343 


1494 


2036 


2138 


2729 


2781 


3421 


3423 


4112 


4065 


4802 


4706 


5491 


5345 


6180 


5985 


6867 


6623 


7554 


7260 


8240 


7897 


8926 


8533 


9610 


9168 


•972 0294 


9802 


0976 


•976 0435 


1658 


1067 


2339 


1699 


3020 


2330 


3699 


2960 


4378 


3589 


5056 


4217 


5733 


4845 


6409 


5472 


7084 


6098 


7759 


6723 


8432 


7347 


9105 


7970 


9777 


8593 


•9730449 


9215 


1119 


9836 


1789 
2458 


•977 0456 
1075 



3125 
3793 
4458 
5124 
5789 
6453 
7116 
7778 
8439 
9100 
9760 
•9740419 
1077 
1734 
2390 
3046 
3701 

13° 



1693 
2311 
2928 
3544 
4159 
4773 j 
5386 | 
5999 
6611 
7222 
7832 
8441 
9050 
9658 
•978 0265 I 
0871 
14761 
12° I 



78° 1 


79° 


978 1476 


•981 6272 


2080 


6826 


2684 


7380 


3287 


7933 


3889 


8485 


4490 


9037 


5090 


9587 


5689 


•982 0137 


6288 


0686 


6886 


1234 


7483 


1781 


8079 


2327 


8674 


2873 


9268 


3417 


9862 


3961 


979 0455 


4504 


1047 


5046 


1638 


5587 


2228 


6128 


2818 


6668 


3406 


7206 


3994 


7744 


4581 


8282 


5167 


8818 


5752 


9353 


5337 


9888 


6921 


•983 0422 


7504 


0955 


8086 


1487 


8667 


2019 


9247 


2549 


9827 


3079 


•980 0405 


3608 


0983 


4136 


1560 


4663 


2136 


5189 


2712 


5715 


3286 


6239 


3860 


6763 


4433 


7286 


5005 


7808 


5576 


8330 


6147 


8850 


6716 


9370 


7285 


9889 


7853 


•984 0407 


8420 


0924 


8986 


1441 


9552 


1956 


•981 0116 


2471 


0680 


2985 


1243 


3498 



1805 
2366 
2927 
3486 
4045 
4603 
5160 
5716 
6272 
11° 



4010 
4521 
5032 
5542 
6050 
6558 
7066 
7572 
8078 

10° 



80° 


81° 


9848 078 


•9876 883 


582 


•9877 338 


•9849 086 


792 


589 


•9878 245 


9850 091 


697 


593 


•9879 148 


9851 093 


599 


593 


•9880 048 


•9852 092 


497 


590 


945 



9853 087 
583 

9854 079 
574 

•9855 068 

561 
■9856 053 

544 
•9857 035 

524 
•9858 013 

501 

988 
•9859 475 

960 
•9860 445 

929 
•9861 412 

894 
•9862 375 

856 
•9863 336 

815 
•9864 293 

770 
•9865 246 

722 
•9866 196 

670 
•9867 143 

615 
•9868 087 

557 
•9869 027 

496 

964 
•9870 431 

897 
•9871 363 

827 
•9872 291 

754 
•9873 216 

678 
•9874 138 

598 
•9875 057 

514 

972 
•9876428 

883 

9° 



•9881 392 

838 
•9882284 

728 
•9883172 

615 
•9884 057 

498 

939 
•9885 378 

817 
•9886 255 

692 
•9887 128 

564 

998 
•9888 432 

865 
•9889 297 

728 
•9890 159 

588 
•9891 017 

445 

872 
••9892 298 

723 
•9893 148 

572 

994 
•9894416 

838 
•9895 258 

677 
•9896 096 

514 

931 
•9897 347 

762 
•9898 177 

590 
•9899 003 

415 

820 
•9900 237 

646 
•9901 055 

4C2 

869 
•9902 275 

681 

8° 



82° 


/ 


9902 681 


60 


9903 085 


59 


489 


58 


891 


57 


•9904 293 


56 


694 


55 



NAT. COSINE 



NATURAL SIKES. 



255 



/ 


83° 


84° 


85° 


86° 


87° 


88° 


89° 


/ 





•9925 462 


•9945 219 


•9961 947 


•9975 641 


•9986 295 


9993 908 


•9998 477 


60 


1 


816 


523 


•9962 200 


843 


447 


•9994 009 


527 


59 


2 


•9926169 


825 


452 


•9976 045 


598 


110 


577 


58 


3 


521 


•9946 127 


704 


245 


748 


209 


625 


57 


4 


873 


428 


954 


445 


898 


308 


673 


56 


5 


•9927 224 


729 


•9963 204 


645 


•9987 046 


405 


720 


55 


6 


573 


•9947 028 


453 


843 


194 


502 


766 


54 


7 


922 


327 


701 


•9977 040 


340 


698 


812 


53 


8 


•9928 271 


625 


948 


237 


486 


693 


856 


52 


9 


618 


921 


•9964 195 


433 


*• 631 


788 


900 


61 


10 


965 


•9948 217 


440 


627 


775 


881 


942 


50 


11 


•9929 310 


513 


685 


821 


919 


974 


984 


49 


12 


655 


807 


929 


•9978 015 


•9988 061 


•9995 066 


•9999 025 


48 


13 


999 


•9949 101 


•9965172 


207 


203 


157 


065 


47 


14 


•9930 342 


393 


414 


399 


344 


247 


105 


46 


15 


685 


685 


655 


589 


484 


336 


143 


45 


16 


•9931 026 


976 


895 


779 


623 


424 


161 


44 


17 


367 


•9950 266 


•9966135 


968 


761 


512 


218 


43 


18 


706 


556 


374 


•9979 156 


899 


599 


254 


42 


19 


•9932 045 


844 


612 


343 


•9989 035 


684 


289 


41 


20 


384 


•9951 132 


849 


530 


171 


770 


323 


40 


21 


721 


419 


•9967 085 


716 


306 


854 


357 


39 


22 


•9933 057 


705 


321 


900 


440 


937 


389 


38 


23 


393 


990 


555 


•9980 084 


573 


•9996 020 


421 


37 


24 


728 


•9952 274 


789 


267 


706 


101 


452 


36 


25 


•9934 062 


557 


•9968 022 


450 


837 


182 


482 


35 


26 


395 


840 


254 


631 


968 


262 


511 


34 


27 


727 


•9953 122 


485 


811 


•9990 098 


341 


539 


33 


28 


•9935 058 


403 


715 


991 


227 


419 


567 


32 


29 


389 


683 


945 


•9981 170 


355 


497 


593 


31 


30 


719 


962 


•9969 173 


348 


482 


573 


619 


30 


31 


•9936 047 


•9954240 


401 


525 


609 


649 


644 


29 


32 


375 


517 


628 


701 


734 


724 


668 


28 


33 


703 


794 


854 


877 


859 


798 


692 


27 


34 


•9937 029 


•9955 070 


•9970 080 


•9982052 


983 


871 


714 


26 


35 


355 


345 


304 


225 


•9991 106 


943 


736 


25 


36 


679 


620 


528 


398 


228 


•9997 015 


756 


24 


37 


•9938 003 


893 


750 


" 570 


350 


086 


776 


23 


38 


326 


•9956 165 


972 


742 


470 


156 


795 


22 


39 


648 


437 


•9971 193 


912 


590 


224 


813 


21 


40 


969 


708 


413 


•9983 082 


709 


292 


831 


20 


41 


•9939 290 


978 


633 


250 


827 


360 


847 


19 


42 1 610 


•9957 247 


851 


418 


944 


426 


863 


18 


43 


928 


515 


•9972 069 


585 


•9992 060 


492 


878 


17 


44 


•9940 246 


783 


286 


751 


176 


556 


892 


16 


45 


563 


•9958 049 


502 


917 


290 


620 


905 


15 


46 


880 


315 


717 


•9984081 


404 


683 


917 


14 


47 ) -9941 195 


580 


931 


245 


517 


745 


928 


13 


48 


510 


844 


•9973 145 


408 


629 


807 


939 


12 


49 


823 


•9959 107 


357 


570 


740 


867 


949 


11 


50 


•9942136 


370 


569 


731 


851 


927 


958 


10 


51 


448 


631 


• 780 


891 


960 


986 


966 


9 


52 


760 


892 


990 


•9985 050 


•9993 069 


•9998 044 


973 


8 


53 


•9943 070 


-9960 152 


•9974199 


209 


177 


101 


979 


7 


54 


379 


411 


408 


367 


284 


157 


985 


6 


55 


688 


669 


615 


524 


390 


213 


989 


5 


56 


996 


926 


822 


680 


495 


267 


993 


4 


57 


•9944 303 


•9961 183 


•9975 028 


835 


600 


321 


996 


3 


58 


609 


438 


233 


989 


704 


374 


998 


2 


59 


914 


693 


437 


•9986143 


806 


426 


1*0000 000 


1 


60 


•9945 219 


947 


641 


295 


908 


477 


000 








1 6° 


I 6° 1 4° 


3° 


2° 


1° 


0° 


! ' 








N 


AT. COSI] 


SE. 









256 



NATURAL TANGENTS. 



0° 

•000 0000 
2909 
5818 
8727 

•001 1636 

- 4544 

7453 

•0020362 
3271 
6180 
9089 

•003 1998 
4907 
7816 

004 0725 
3634 
6542 
9451 

•005 2360 
5269 

81' 

006 1087 
3996 
6905 
9814 

007 2723 
5632 
8541 

008 1450 
4360 
7269 

009 017 
3087 
5996 
8905 

•010 1814 
4724 
7633 

Oil 0542 
3451 
6361 
9270 

012 2179 
5088 
7998 

013 0907 
381' 
6726 
9635 

014 2545 
5454 
8364 

015 1273 
4183 
7093 

0160002 
2912 
5821 
8731 

017 1641 

4551 

89< 



017 4551 
7460' 

018 0370 
3280 
6190 
9100 

•019 2010 
4920 
7830 

•020 0740 
3650 
6560 
9470 

•021 2380 
5291 
8201 

•022 1111 
4021 
6932 
9842 

023. 2753 
5663 
8574 

•024 1484 
4395 
7305 

•025 0216 
3127 
6038 
8948 

•026 1859 
4770 
7681 

•027 0592 
3503 
6414 
9325 

•028 2236 
5148 
8059 

•029 0970 
3882 
6793 
9705 

030 2616 
5528 
8439 

031 1351 
4263 
7174 

032 0086 
2998 
5910 
8822 

033 1734 
4646 
7558 

034 0471 
3383! 
6295' 
9208! 



2° 

034 9208 

035 2120 
5033 
7945 

•0360858 
3771 
6683 
9596 

•037 2500 
5422 
8335 

•038 1248 
4161 
7074 
9988 

•039 2901 
5814 
8728 

•040 1641 
4555! 
7469 1 

•041 0383; 
3296! 
6210; 
9124 

•042 2038; 
4952' 
7866' 

•043 0781 
3695 
6609! 
9524| 

•044 2438! 
5353 
8268! 

•045 1183 
40971 
7012' 
9927 ! 

•046 2842 

1 5757 
8673 

•047 1588 
4503 
7419! 

•048 0334 
3250] 
6166' 
9082 

•049 1997 1 
4913' 
7829| 

•050 0746 
3662; 
6578- 
9495| 

•051 2411| 
5328, • 
8244 

052 1161 
4078 

87° 



3° 


4° 


052 4078 


•069 9268 • 


6995 


•070 2191 


9912 


5115- 


053 2829 


8038 


5746 


•071 0961 


8663 


3885 


054 1581 


6809 


4498 


9733 


7416 


•072 2657 


•055 0333 


5581 


3251 


8505 


6169 


073 1430 


9087 


4354 


•056 2005 


7279 


4923 


•0740203 


7841 


3128 


•057 0759 


6053 


3678 


8979 


6596 


•075 1904 


9515 


4829 


•058 2434 


7755 


5352 


•076 0680 


8271 


3606 


•059 1190 


6532 


4109 


9458 


7029 


•077 2384 


9948 


5311 


•060 2867 


8237 


5787 


•078 1164 


8706 


4090 


•061 1626 


7017 


4546 


9944 


7466 


•079 2871 


•062 0386 


5798 


3306 


8726 


6226 


•080 1653 


9147 


4581 


•063 2067 


7509 


4988 


•081 0437 


7908 


3365 


•064 0829 


6293 


3750 


9221 


6671 


•082 2150 


9592 


^ 5078 


•065 2513 


8007 


5435 


•083 0936 


8356 


G865 


•066 1278 


6794 


4199 


9723 


7121 


•084 2653 


•067 0043 


5583 


2965 


8512 


5887 


•085 1442 


8809 


4372 


068 1732 


73021 


4654 


•086 0233J 


7577 


3163 


069 0499 


6094!' 


3422 


9025! 


6345 


•087 1956 


9268 


4887- 


86° 


85° 1 



•087 4887 
7818 

088 0749 
3681 
6612 
9544 

089 2476 
5408 
8341 

090 1273 
4200 
7138 

091 0071 
3004 
5938 
8871 

092 1804 
4738 
7672 

093 0606 
3540 
6474 
9409 

•094 2344 
5278 
8213 

•095 1148 
4084 
7019 
9955 

•096 2890 
5826 
8763 

•097 1699 
4635 
7572 

•098 0509 
3446 
6383 
9320 

•099 2257 
5194 
8133 

•100 1071 
4009 
6947 
9886 

•101 2824 
5763 
8702 

102 1641 
4580 
7520 

103 0460 
3399 
6340 
9280 

104 2220 
5161 
8101 

105 1042 
84° 



6° 

105 1042 
3983 
6925 
9866 

106 2808 
5750 
8692 

107 1634 
4576 
7519 

108 0462 
3405 
6348 
9291 

109 2234 
5178 
8122 

110 1060 
4010 
6955 
9899 

•111 2844 
5789 
8734 

•112 1680 
4625 
7571 

•113 0517 
3463 
6410 
9356 

•114 2303 
5250 
819' 

•115 1144 
4092 
7039 
9987 

•116 2936 
5884 
8832 

•117 1781 
4730 
7679 

•118 0628 
3578 
6528 
9478 

119 2428 
5378 
8329 

120 1279 
4230 
7182 

121 0133 



122 1941 
4893 
7846 
83° 



7° 

122 7846 

123 0798 
3752 
6705 
9658 

124 2612 
5566 
8520 

125 1474 
4429 
7384 

126 0339 
3294 
6249 
9205 

•127 2161 
5117 
8073 

■128 1030 
3986 
6943 
9900 

•129 2858 
5815 
8773 

•130 1731 
4690 
7648 

•131 0607 
3566 

6525 
9484 

'132 2444 
5404 
8364 

•1331324 
4285 
7246 

•134 0207 
3168 
6129 
9091 

•135 2053 
5015 
7978 

•136 0940 
3903 
6866 
9830 

137 2793 

5757 
8721 
138 1685 
4650 
7615 

139 0580 
3545 
6510 
9476 

140 2442 
5408 

82* 



NAT. 0OTA.N. 



NATURAL TANGENTS. 



257 



8° 


9° I 


10° 


11° 


12° ! 


13° 1 


14° 1 


15° 1 


/ 


•140 5408 


158 38441.176 3270 


194 3803 


212 556ti -230 8682 


249 3280 


257 9492 


60 


8375 


68261 6269 


6822 


8606 -231 1746 


6370 


268 2610 


59 


•1411342 


9809 | 9209 


9841 


2131647 4811 


9460 


5728 


58 


4308 


159 2791 -177 2269 


195 2861 


4688 7876- 


250 2551 


8847 


57 


7276 


5774 


5270 


5881 


7730 -232 0941 


5642 


269 1967 


56 


•142 0243 


8757 


8270 


8901 


214 0772 4007 


87341 


5087 


55 


3211 


160 1740 


178 1271 


196 1922 


3814' 7073 


251 1826 


8207 


54 


6179 


4724 


4273 


4943 


6857 


233 0140 


4919 


270 1328 


53 


9147 


7708 


7274 


7964 


9900 


3207 


8012 


4449 


52 


•143 2115 


161 0692 


179 0276 


197 0986 


215 2944 


6274 


252 1106 


7571 


51 


5084 


3677 


3279 


4008 


5988 


9342 


4200 


271 0694 


50 


8053 


6662 


6281 


7031 


9032 


234 2410 


7294 


3817 


49 


•144 1022 


9647 


9284 


198 0053 


216 2077 


5479 


253 0389 


6940 


48 


3991 


162 2632 -180 2287 


3076 


5122 


8548 


3484 


•272 0064 


47 


6961 


5618 


5291 


6100 


8167 


•235 1617 


6580 


3188 


46 


9931 


8603 


8295 


9124 


•217 1213 


4687 


9676 


6313 


45 


•145 2901 


163 1590 


181 1299 


199 2148 


4259 


7758 


254 2773 


9438 


44 


5872 


4576 


4303 


5172 


7306 


•236 0829 


5870 


•273 2564 


43 


8842 


7563 


7308 


8197 


•218 0353 


3900 


8968 


5690 


42 


•146 1813 


•164 0550 182 0313 


•200 1222 


3400 


6971 


•255 2066 


8817 


41 


4784 


3537 


3319 


4248 


6448 


.237 0044 


5165 -274 1945 


40 


7756 


6525 


6324 


7274 


9496 


3116 


8264 


5072 


39 


•147 0727 


9513 


9330 


•201 0300 


•219 2544 


6189 


•256 1363 


8201 


38 


3699 


•165 2501 


•183 2337 


3327 


5593 


9262 


4463 -275 1330 


37 


6672 


5489 


5343 


6354 


8643 


•238 2336 


7564 4459 


36 


9644 


8478 


8350 


9381 


•220 1692 


5410 


•257 0664 7589 


35 


•148 2617 


•166 1467 


•1841358 


•202 2409 


4742 


8485 


3766 


•276 0719 


34 


5590 


4456 


4365 


5437 


7793 


•239 1560 


6868 


3850 


33 


8563 


7446 


7373 


8465 


•221 0844 


4635 


9970 


6981 


32 


•149 1536 


•167 0436 


•185 0382 


•203 1494 


3895 


7711 


•258 3073 


•277 0113 


31 


4510 


3426 


3390 


4523 


6947 


•2400788 


6176 


3245 


30 


7484 


6417 


6399 


7552 


9999 


3864 


9280 


6378 


29 


•150 0458 


9407 


9409 


•204 0582 


•222 3051 


6942 


•259 2384 


9512 


28 


3433 


•168 2398 j-186 2418 


3612 


6104 


•241 0019 


5488 


•278 2646 


27 


6408 


5390 5428 


6643 


9157 


3097 


8593 


5780 


26 


9383 


8381 8439 


9674 


•223 2211 


6176 


•260 1699 


8915 


25 


•151 2358 


•169 1373 -187 1449 


•205 2705 


5265 


9255 


4805 


•279 2050 


24 


5333 


4366 


4460 


5737 


8319 


•242 2334 


7911 


5186 


23 


8309 


7358 


7471 


8769 


•224 1374 


5414 


•261 1018 


8322 


22 


•1521285 


•170 0351 


•188 0483 


•206 1801 


4429 


8494 


4126 


•280 1459 


21 


4262 


3344 


3495 


4834 


7485 


•2431575 


7234 


4597 


20 


7238 


6338 


6507 


7867 


•225 0541 


4656 


•262 0342 


7735 


19 


•153 0215 


9331 


9520 


•207 0900 


3597 


7737 


3451 


•281 0873 


18 


3192 


•171 2325 


•189 2533 


3934 


6654 


•244 0819 


6560 


4012 


17 


6170 


5320 


5546 


6968 


9711 


3902 


9670 


7152 


16 


9147 


8314 


8559 


•208 0003 


•226 2769 


6984 


•263 2780 


•282 0292 


15 


154 2125 


•172 1309 -190 1573 


3038 


5827 


•245 0068 


5891 


3432 


14 


5103 


4304 4587 


6073 


8885 


3151 


9002 


6573 


13 


8082 


7300 


7602 


9109 


•227 1944 


6236 


•264 2114 


9715 


12 


•155 1061 


173 0296 


•191 0617 


•209 2145 


5003 


9320 


5226 


•283 2857 


11 


4040 


3292 


3632 


5181 


8063 


•246 2405 


8339 


5999 


10 


7019 


6288 


6648 


8218 


•228 1123 


5491 


•265 1452 


9143 


9 


9998 


9285 


9664 


•210 1255 


4184 


8577 


4566 


•284 2286 


8 


•156 2978 


•174 2282 


•192 2680 


4292 


7244 


•247 1663 


768C 


5430 


7 


5958 


5279 


5696 


7331 


•229 0306 


4750 


•266 0794 


8575 


6 


893S 


8277 


8713 


•211 036£ 


3367 


7837 


3909 


•285172C 


5 


•157 1912 


•175 1275 


•193 1731 


3407 


6429 


•24* 0925 


7025 


4866 


4 


490C 


4273 


4748 


644C 


9495 


4013 -267 0141 


8012 


3 


7881 


7272 


7766 


948f 


•230 2555 


7102 


3257 


•2861159 


2 


•158 0865 


•176 0271 


•1940784 


•212 252,: 


5615 


•249 0191 


6374 


430C 


1 


3844 


327C 


3802 


556^ 


> 8685 


\ 328C 


9492 


7454 





81° 


80° 


79° 


78° 
NAT. 


77° 

COTAN. 


76° 


75° 


74° 


' 



258 



NATURAL TANGENTS. 



16° 1 


17° 


18° 


19° 


20° 


21° 


22° 


23° 


•286 7454 


305 7307 


324 9197 


344 3276 


363 9702 


383 8640 


404 0262 


•424 4748 


•287 0602 


306 0488 


325 2413 


6530 


364 2997 


384 1978 


3646 


8182 


3751 


3670 


5630 


9785 


629^ 


5317 


7031 


•425 1616 


6900 


6852 


8848 


345 3040 


9588 


8656 


405 0417 


5051 


•288 0050 


307 0034 


326 2060 


6296 


365 2885 


385 1996 


3804 


8487 


3201 


3218 


5284 


9553 


6182 


5337 


7191 


•426 1924 


6352 


6402 


8504 


346 2810 


9480 


8679 


4060579 


5361 


9503 


9586 


327 1724 


6068 


•366 2779 


386 2021 


3968 


8800 


•289 2655 


308 2771 


4944 


9327 


6079 


5364 


7358 


•427 2239 


5808 


5957 


8165 


347 2586 


9379 


8708 


407 0748 


5680 


8961 


9143 


328 1387 


5846 


■367 2680 


387 2063 


4139 


9121 


•290 2114 


•30C 2330 


4610 


9107 


5981 


5398 


7531 


•428 2563 


5269 


5517 


7833 


348 2368 


9284 


8744 


408 0924 


6005 


8423 


8705 


329 1056 


5630 


•368 2587 


388 2091 


4318 


9449 


•291 1578 


•310 1893 


4281 


8893 


5890 


5439 


7713 


•429 2894 


4734 


5083 


7505 


349 2156 


9195 


8787 


•409 1108 


6339 


7890 


8272 


330 0731 


5420 


•369 2500 


•389 2136 


4504 


9785 


•292 1047 


•311 1462 


3957 


8685 


580t 


5486 


7901 


•430 3232 


4205 


4653 


7184 


350 1950 


9112 


8837 


•410 1299 


6680 


7363 


7845 


•331 0411 


5216 


•370 2420 


•390 2189 


4697 


-431 O* 29 


•293 0521 


•312 1036 


3639 


8483 


5728 


5541 


8097 


3579 


3680 


4229 


6868 


•351 1750 


9036 


8894 


•411 1497 


7030 


6839 


7422 


•332 0097 


5018 


•371 2346 


•391 2247 


4898 


•432 0481 


9999 


•313 0616 


3327 


8287 


5656 


5602 


8300 


3933 


•294 3160 


3810 


6557 


•352 1556 


8967 


8957 


•412 1703 


7386 


6321 


7005 


9788 


4826 


•372 2278 


•392 2313 


5106 


•433 0840 


9483 


•3140200 


•3333020 


8096 


5590 


5670 


8510 


4295 


•295 2645 


3396 


6252 


.0*0 1368 
^4640 


8903 


9027 


•4131915 


7751 


5808 


6593 


9485 


.373 2217 


•393 2386 


5321 


.434 1208 


8971 


9790 


•334 2719 


7912 


5532 


5745 


8728 


4665 


•296 2135 


•315 2988 


5953 


•3541186 


8847 


9105 


•414 2136 


8124 


5299 


6186 


9188 


4460 


•374 2163 


•394 2465 


5544 


•435 1583 


8464 


9385 


•335 2424 


7734 


5479 


5827 


8953 


5043 


•297 1630 


•316 2585 


5660 


•355 1010 


8797 


9189 


•415 2363 


8504 


4796 


5785 


8896 


4286 


•375 2115 


•395 2552 


5774 


•4361966 


7962 


8986 


•3362134 


7562 


5433 


5916 


9186 


5429 


•298 1129 


•317 2187 


5372 


•356 0840 


8753 


9280 


•416 2598 


8893 


4297 


5389 


8610 


4118 


•376 2073 


•396 2C45 


6012 


•437 2357 


7465 


8591 


•337 1850 


7397 


5394 


6011 


9426 


5823 


•299 0634 


•318 1794 


5090 


•357 0676 


8716 


9378 


•417 2841 


9289 


3803 


4998 


8330 


3956 


•377 2038 


•397 2746 


6257 


•438 2756 


6973 


8202 


•338 1571 


7237 


5361 


6114 


9673 


6224 


•300 0144 


•319 1407 


4813 


•358 0518 


8685 


9483 


•418 3091 


9693 


3315 


4613 


8056 


3801 


•378 2010 


•398 2853 


6509 


•439 3163 


6486 


7819 


•3391299 


7083 


5335 


6224 


9928 


6634 


9658 


•320 1025 


4543 


•359 0367 


8661 


9595 


•419 3348 


•440 0105 


•301 2831 


4232 


7787 


3651 


•379 1988 


•399 2968 


6769 


3578 


6004 


7440 


•340 1032 


6936 


5315 


6341 


•420 0190 


7051 


9178 


•321 0649 


4278 


•360 0222 


8644 


9715 


3613 


•441 0526 


•302 2352 


3858 


7524 


3508 


•380 1973 


•400 3089 


7036 


4001 


5527 


7067 


.341 0771 


6795 


5302 


6465 


•421 04C0 


7477 


8703 


•3220278 


4019 


•361 0082 


8633 


9841 


3885 


4420954 


•303 1879 


3489 


7267 


3371 


•381 1964 


.401 3218 


7311 


4432 


5055 


6700 


•342 0516 


6660 


5296 


6596 


•422 0738 


7910 


8232 


9912 


3765 


9949 


8629 


9974 


4165 


•4431390 


•3041410 


•323 3125 


7015 


•362 3240 


•382 1962 


•4023354 


7594 


4871 


4588 


6338 


•343 0266 


6531 


5296 


6734 


v J3 1023 


8352 


7767 


9552 


3518 


9822 


8631 


•403 0115 


4453 


•4441834 


•305 0946 


•324 2766 


6770 


•363 3115 


•383 1967 


3496 


7884 


5318 


4126 


5981 


•344 0023 


6408 


5303 


6879 


•4241316 


8802 


7307 


9197 


3276 


9702 


8640 


•404 0262 


4748 


•445 2287 


73° 


72° 


71° 


70° 

HAT. < 


69° 
30TAN 


68° 


67° 


66° 



NATURAL TANGENTS. 



259 



24° 

•445 2287 
5773 
9260 

446 2747 
6236 
9726 

•447 3210! 
67081 

•448 0200 ' 
3693 1 
7187! 

•449 0682; 
4178: 
7675 

•4501173! 
4672 1 
8171 

•4511672! 
5173! 
8676! 

452 2179 
5683 
9188 

•453 2694 
6201 
9709 

•4543218 
6728 

•455 0238 
3750 
7263 

•456 0776 
4290 
7806 

•457 1322 



25° 

•466 3077 
6618 

467 0161 
3705 
7250 

468 0796 
4342 
7890 

469 1439 
4988 
8539 

470 2090 
5643 
9196 

471 2751 
6306 
9863 

472 3420 
6978 

473 0538 
4098 
7659 

■474 1222 
4785 
8349 

•4751914 
5481 
9048 

476 2616 
6185 
9755 

477 3326 



•478 0472 

4046 

7821 

8357 '"479 119' 

•4581877| 4774 

5397! 8352 

8918-4801932 



•459 2439! 
5962 
9486; 

•460 3011i 
6537! 

•4610063'' 
35911 
7119; 

•462 0649! 
4179. 

7710! 
•463 1243 ! 

4776 : 

8310; 
•4641845! 

5382 

8919' 
•465 2457! 

5996! 

9536; 
•466 30771 

65° 



5512 
9093 

481 2675 
6258 
9842 

482 3427 
7014 

483 0601 
4189 
7778 

434 1368 
4959 
8552 

■485 2145| 
5739, 
9334 

■486 2931 1 
6528! 

487 0126' 
3726' 
7326 
64° 



26' 

•487 7326 

488 092' 
4530 
8133 

■489 173 
5343 
8949 

■490 2557 
6166 
9775 

'491 3386 
6997 

•4920610 
4224 
7838| 

'493 1454 1 
50711 
8689! 

'494 2308! 
59281 
9549! 

■495 3171] 
6794! 

•496 0418; 
4043; 
7669| 

■497 1297 
4925 
8554 

■498 2185 
5816 
9449 

•499 3082 
6717 

■500 0352 
3989 
7627 

•501 1266 
4906 
8547 

•502 2189!' 
5832| 
9476! 

■503 31211 
6768; 

•5040415' 
4063 J 
7713: 

■505 1363 
5015i 



506 2322 
5977' 
9633 

507 3290 
6948 

508 0607 ' 
4267 
7929 

509 1591 

5254 

63° [ 



27° ! 

509 5254 
8919 

510 2585 
6252 
9919 

511 3588 
7259 

5120930 
4602 
8275 

513 1950 
5625 
9302 

514 2980 
6658 

'•515 0338 
4019 
7702 

516 1385 
5069 
8755 

517 2441 
6129 
9818 

■518 3508 
7199 

•519 0891 
4584 
82' 

520 1974 

5671 
9368 

■521 3067 
676: 

■522 0468 
4170 
7874 

5231578 
5284 
8990 

■5242698 
6407 

525 0117 
3829 
7541 

5261255 
4969 
8685 

■527 2402 
6120 
9839 

528 3560 
7281 

529 1004 
4727 
8452 

530 2178 
5906 
9634 

531 3364 
7094 

62' 



o 



28' 

531 7094 

532 0826 
4559 
8293 

533 2029 
5765 
9503 

534 3242 
6981 

535 0723 
4465 
8208 

536 1953 
5699 
9446 

537 3194 
6943 

538 0694 
4445 
8198 

539 1952 
5707 
9464 

•540 3221 
6980 

•541 0740 
4501 
8263 

•542 20! 
5791 
955' 

•543 3324 
7092 

•544 0862 
4632 
8404 

-545 217 
5951 
9727 

•546 3503 
7281 

•547 1060 
4840 
8621 

•548 2404 
6188 
9973 

•549 3759 
7547 

•550 1335 
5125 
8916 

•551 2708 
6502 

•552 0297 
4093 
7890 

•5531688 
5488 
9288 

•554 30911 

61° 



29° 1 

554 3091 
6894 

555 0698 
4504 
8311' 

556 2119 
5929 
9739 

557 3551) 
7364! 

■558 1179 
4994 1 
8811 

■559 2629 
6449 

•560 0269 
4091 
7914 

•561 1738 
5564 
9391 

•562 3219 
7048 

•563 0879 
4710 
8543 

■564 2378 
6213 

•565 0050 
3888 
7728 

•566 1568 
5410 
9254 

■567 3098 
6944 

■568 0791 
4639 
8488 

•569 2339 
6191 

■570 0045 
3899 
7755 

•571 1612 
5471 
9331 

•572 3192 
7054 

•573 0918 
4783 
8649 

•574 2516 
6385 

•575 0255 
4126 
7999 

■5761873 
5748 
9625 

•577 3503 
60° 



30 c 

577 3503 

7382 

578 1262 
5144 
902^ 

579 2912 
6797 

580 0684 
4573 
8462 

581 2353 
6245 

582 0139 
4034 
7930 

583 1828 
5726 
9627 

584 3528 
7431 

585 1335 
5241 
9148 

586 3056 
6965 

587 0876 
4788 
8702 

588 2616 
6533 

589 0450 
4369 
8289 

590 2211 
6134 

591 0058 
3984 
7910 

•592 1839 
5768 
9699 

593 3632 
7565 

5941501 
5437 
9375 

595 3314 
7255 

596 1196 
5140 
9084 

597 3030 
6978 

598 0926 
4877 
8828 

599 2781 
6735| 

600 0691, 
4648 
8606 

59° I 



31° I > 

600 8606 60 

•6012566 59 

6527 58 

•602 0490 57 

4454 56 

8419 55 

•603 2386 54 

6354 53 

•604 0323 52 

4294 51 

8266 1 50 

•605 2240 49 

6215 48 

606 0192 47 

4170 46 

8149 45 

•607 2130 44 

6112 43 

608 0095: 42 

4080 ( 41 

8067 1 40 
•609 2054' 39 

6043 38 
•610 0034 37 

4026, 36 

8019 35 
6112014' 34 

6011 33 
•612 0008 ' 32 

4007 j 31 

8008! 30 
613 2010,' 29 

6013! 28 
6140018! 27 

4024J 26 

8032 25 

615 2041 24 
6052 23 

616 0064 22 
4077 21 
8092 20 

617 2108; 19 
01 26| 18 

618 0145J 17 
4166 16 
8188 15 

619 22111 14 
6236, 13 

620 0263 12 
4291 11 

10 
9 

8 
7 
6 
5 
4 



8320 

621 2351 
6383 

622 0417 
4452 
8488 

623 2527 
6566 

624 0607 
4650! 
8694 

68° 



NAT. COTAN. 



260 



NATURAL TANGENTS. 



32° 


33° 


34° 


35° 


36° 


'37° 1 


38° 1 


39° 


624 8694 


•649 4076 


•674 5085 


•700 2075 


■726 5425 


753 5541 


•781 28561-809 7840 


625 2739 


8212 


9318 


6411 


9871 


754 0102 


7542 


•810 2658 


6786 


•650 2350 


•675 3553 


•701 0749 


•727 4318 


4666 


•782 2229 


7478 


626 0834 


6490 


7790 


5089 


8767 


9232 


6919 


•811 2300 


4884 


•651 0631 


•676 2028 


9430 


•728 3218 


•755 3799 


•7831611 


7124 


8935 


4774 


6268 


•702 3773 


7671 


8369 


6305 


•8121951 


627 2988 


8918 


•677 0509 


8118 


•729 2125 


•756 2941 


•784 1002 


6780 


7042 


•652 3064 


4752 


•703 2464 


6582 


7514 


5700 


•813 1611 


628 1098 


7211 


8997 


6813 


•730 1041 


•757 2090 


•785 0400 


6444 


5155 


•653 1360 


•678 3243 


•704 1163 


5501 


6668 


5103 


•8141280 


9214 


5511 


7492 


5515 


9963 


•758 1248 


9808 


6118 


629 3274 


9663 


•679 1741 


9869 


•731 4428 


5829 


•786 4515 


•815 0958 


7336 


•654 3817 


5993 


•705 4224 


8894 


•759 0413 


9224 


5801 


630 1399 


7972 


•680 0246 


8581 


•732 3362 


4999 


•787 3935 


•816 0646 


5464 


•655.2129 


4501 


•7062940 


7832 


9587 


8649 


5493 


9530 


6287 


8758 


7301 


•733 2303 


•760 4177 


•788 3364 


•817 0343 


631 3598 


•656 0447 


•681 3016 


•707 1664 


6777 


8769 


8082 


5195 


7667 


4609 


7276 


6028 


•7341253 


•761 3363 


•789 2802 


•818 0049 


632 1738 


8772 


•6821537 


•708 0395 


5730 


7959 


7524 


4905 


5810 


•657 2937 


5801 


4763 


•735 0210 


•762 2557 


•790 2248 


9764 


9883 


7103 


•683 0066 


9133 


4691 


7157 


6975 


•819 4625 


633 3959 


•658 1271 


4333 


•709 3504 


9174 


•7631759 


•7911703 


9488 


8035 


5441 


8601 


7878 


•736 3660 


6363 


6434 


•820 4354 


634 2113 


9612 


•684 2871 


•710 2253 


8147 


•764 0969 


•792 1167 


9222 


6193 


•659 3785 


7143 


663C 


•737 2636 


5577 


5902 


•821 4093 


.635 0274 


7960 


•6851416 


•711 1009 


7127 


•765 0188 


•793 0640 


8965 


4357 


•660 2136 


5692 


539C 


•738 1620 


4800 


5379 


•822 3840 


8441 


6313 


9969 


9772 


6115 


9414 


•794 0121 


8718 


•636 2527 


•661 0492 


•686 4247 


•712 4157 


739 0611 


•766 4031 


4865 


.823 3597 


6614 


4673 


8528 


8543 


5110 


8649 


9611 


8479 


637 0703 


8856 


•687 2810 


•713 2931 


9611 


•767 3270 


•795 4359 


•824 3364 


4793 


•662 3040 


7093 


7320 


•740 4113 


7893 


9110 


8251 


8885 


7225 


•688 1379 


•7141712 


8618 


•768 2517 


•796 3862 


•825 3140 


•638 2978 


•6631413 


5666 


6ioe 


•741 3124 


7144 


8617 


8031 


7073 


5601 


9955 


•715 0501 


7633 


•769 1773 


•797 3374 


•826 2925 


•639 1169 


9792 


•689 4246 


4898 


•742 2143 


6404 


8134 


7821 


5267 


•664 3984 


8538 


9297 


6655 


•770 1037 


•798 2895 


•827 2719 


9366 


8178 


•690 2832 


•716 3698 


•743 1170 


5672 


7659 


7620 


•640 3467 


•665 2373 


7128 


8100 


5686 


•771 0309 


•799 2425 


•828 2523 


7589 


6570 


•691 1425 


•717 2505 


•744 0204 


4948 


7193 


7429 


•641 1673 


•666 0769 


5725 


6911 


4724 


9589 


•8001963 


•829 2337 


5779 


4969 


692 0026 


•718 1319 


9246 


•772 4233 


6736 


7247 


9886 


9171 


4328 


5729 


•745 3770 


8878 


•801 1511 


•830 2160 


•642 3994 


•667 3374 


8633 


•719 0141 


8296 


•773 3526 


6288 


7075 


8105 


7580 


.693 2939 


4554 


•746 2824 


8176 


•802 1067 


•831 1992 


•643 2216 


•668 1786 


7247 


8970 


7354 


•774 2827 


5849 


6912 


6329 


5995 


•6941557 


•720 3387 


•747 1886 


7481 


•803 0632 


•832 1834 


.644 0444 


•669 0205 


5868 


7806 


6420 


•775 2137 


5418 


6759 


4560 


4417 


•695 0181 


•721 2227 


•748 0956 


6795 


•804 0206 


•8331686 


8678 


8630 


4496 


6650 


5494 


•7761455 


4997 


6615 


•645 2797 


•670 2845 


881S 


•7221075 


•749 0033 


6118 


9790 


•834 1547 


6918 


7061 


.696 3131 


5502 


4575 


•777 0782 


•805 4584 


6481 


•646 1041 


•671 1280 


7451 


9930 


9119 


5448 


9382 


•835 1418 


5165 


5500 


•697 177S 


■723 4361 


750 3665 


•778 0117 


•806 4181 


6357 


9290 


9721 


6097 


8793 


8212 


4788 


8983 


•836 1298 


•647 3417 


•672 3944 


•698 042:: 


•724 3227 


•751 2762 


9460 


•807 3787 


6242 


7546 


8169 


4749 


7663 


7314 


•779 4135 


8593 


•837 1188 


•648 1676 


•673 2396 


9078 


•725 2101 


•752 1867 


8812 


•808 3401 


6136 


5808 


6624 


•699 3409 


6540 


6423 


•780 3492 


8212 


■838 1087 


9941 


•674 0854 


7741 


•726 0982 


•753 0981 


8173 


•809 3025 


6041 


•649 4076 


5085 


.•700 2075 


5425 


5541 


•781 2856 


7840 


•839 0996 


57° 


56° 


{ 55° 


54° 
NAT. ( 


53° 
20TAN. 


52° 


51° 


60° 



NATURAL TANGENTS. 



261 



40° 


41° 


42° 


839 0996 


•869 2867 


•900 4040 • 


5955 


7976 


9309- 


840 0915 


•870 3087 


•901 4580 


5878 


8200 


9854- 


8410844 


•871 3316 


•902 5131 


5812 


8435 


•903 0411 • 


8420782 


•872 3556 


5C93 


5755 


8680 


•904 0979 • 


8430730 


•873 3806 


6267 


5708 


8935 


•905 1557 • 


•8440688 


•8744067 


6851 


5670 


9201 


•906 2147 • 


•845 0655 


•875 4338 


7446- 


5643 


9478 


•907 2748 


•846 0633 


•876 4620 


8053 • 


5625 


9765 


•908 3360 


•847 0620 


.877 4912 


8671 


5617 


.878 0062 


•909 3984 


•848 0617 


5215 


9300- 


5619 


•879 0370 


•910 4619 


•849 0624 


5528 


9940- 


5631 


•880 0C8S 


•911 5265 • 


•850 0640 


5852 


•912 0592 


5653 


•881 1017 


5922- 


•851 0667 


6186 


•913 1255 


5684 


•882 1357 


6591 • 


•852 0704 


6531 


•914 1929 


5726 


•883 1707 


7270- 


•853 0750 


6886 


•915 2615 


5777 


•884 2068 


7962- 


•8540807 


7253 


•916 3312 


5839 


•885 2440 


8665- 


•855 0873 


.7630 


•917 4020 • 


5910 


•886 2822 


9379 


•856 0950 


8017 


•918 4740 • 


5992 


•887 3215 


•919 0104 


•857 1037 


8415 


5471 


6084 


•888 3619 


•920 0841 


•858 1133 


8825 


6214- 


6185 


•889 4033 


•921 1590 


•859 1240 


9244 


6969 


6297 


•890 4458 


•922 2350 


•860 1357 


9675 


7734 


6419 


•891 4894 


•923 3122 


•8611484 


•892 0116 


8512 


6551 


5341 


•924 3905 


•8621621 


•893 0569 


9301 


6694 


5799 


•9254700 


•8631768 


•8941032 


•926 0102 


6846 


6268 


5506 


•864 1926 


•895 1506 


•927 0914 


7009 


6747 


6324 


•865 2094 


•896 1991 


•928 17 38 


7181 


7238 


7154 


•866 227? 


•897 2487 


•929 2573 


7365 


7739 


7996 


-867 2460 


•898 2994 


•930 3421 


7558 


8251 


8849 


•868 2659 


•899 3512 


•931 4280 


7762 


8775 


9714 


•869 2867 


•900 4040 


•932 5151 


49° 


48° 


47° | 



43° 

93.2 5151 
933 0591 

6034 
934 1479 

6928 

935 2380 
7834 

936 3292 
753 

937 4216 
9683 

938 5153 

939 0625 
6101 

940 1579 
7061 

941 2545 
8033 

9423523 
901' 

943 4513 

944 0013 
5516 

945 1021 
6530 

946 2042 
7556 

947 3074 
8595 

948 4119 
9646 

949 5176 
•950 07< 

6245 

951 1784 

7326 

952 2871 
8420 

953 3971 
9526 

954 5083 

955 0644 
6208 

956 1774 

7344 
957 291' 

8494 
•958 4073 

9655 
•959 5241 
•960 0829 

6421 
•961 2016 

7614 
•962 3215 

8819 
•963 4427 
•964 0037 

5651 
•965 1268 

688 

I 46° 



44° 

•9C5 6888 
•966 2511 

8137 
•967 3767 

9399 
•968 5035 
•969 0674 

6316 
•9701962 

7610 



•971 

•972 
•973 

•974 

•975 

•976 



3262 
891' 
4575 
0236 
5901 
1569 
7240 
2914 
8591 
4272 
9956 

•977 5643 

•978 1333 
702'. 

•979 2724 
8424 

•980 4127 
9833 

•981 5543 

•9821256 
6973 

•983 2692 
8415 

•9844141 
9871 

•985 5603 

•986 1339 
707 

•987 2821 
8567 

•988 4316 

•989 0069 
5825 

•990 1584 
7346 

•991 3112 
8881 

•992 4654 

•993 0429 
6208 

•9941991 
777. 

•995 3566 
9358 

•996 5154 

•997 0953 
6756 

•998 2562 
8371 

.999 4184 
1-000 0000 

45° 



45« 

1-00 00000 
05819 
11642 
17469 
23298 
29131 
34968 
40807 
46651 
52497 
58348 
64201 
70058 
75918 
81782 
87649 
93520 
99394 

1-01 05272 
11153 
17038 
22925 
28817 
34712 
40610 
46512 
52418 
58326 
64239 
70155 
76074 
81997 
87923 
93853 
99786 

1-02 05723 
11664 
17 

23555 
29506 
35461 
41419 
47381 
53346 
59315 
6528 1 
71263 
77243 
83226 
89212 
95203 

1-0301196 
07194 
13195 
19199 
25208 
31220 
37235 
43254 
49277 
55303 

44° 



46° 

1-03 55303 
61333 
67367 
73404 
79445 
85489 
91538 
97589 

1-04 03645 
09704 
1576^ 
21833 
27904 
33977 
40055 
46136 
52221 
58310 
64402 
70498 
76598 
82702 



94920 
1-05 01034 
07153 
13275 
19401 
25531 
31664 
37801 
43942 
5008' 
56235 
62SF.S 
68544 
74704 
80867 
87035 
93206 

99381 
1-0605560 
11742 
17929 
24119 
30313 
36511 
42713 
48918 
65128 
61341 
67558 
73779 
80004 
86233 
92466 
98702 
1-07 04943 
11187 
17435 
23687 

43° 



47° y 

L-07 23687 60 
29943 59 
36203 ! 58 
42467 '57 
48734 56 
55006 55 
61282 54 
67561153 
73845152 
80132 51 
86423 ! 50 
92718149 
99018 48 

1-08 05321 S47 
11628,46 
17939 45 
24254144 
30573.43 
36896 42 
43223 141 
49554 40 
55889 39 
62228,38 
68571 ,37 
74918 '36 
81269 f£ 
87624 34 
9 r >984!33 

1-09 00347132 
06714131 
13085:30 
19460 129 
25840 28 



32223 
38610 
45002 
51397 
57797 
64201 
70609 
77020 
83436 
89857 
96281 
1-10 02709 
09141 
15578 
22019 



27 
26 
25 
24 
23 
22 
21 
20 
19 
IS 
17 
16 
15 
14 
13 



2846312 
3491211 
41365,10 
47823 9 
54284 I 8 
607 50 j 7 
67219 6 
736931 5 
80171 i 4 
866531 3 
93HG 2 
996301 1 
1-11 06125 1 

42° ' 



NAT. COTAN. 



262 



NATURAL TANGENTS. 



U 

e 

7 
8 
9 



48° 

111 06125 
12624 
1&127 
25635 
32146 
38662 
45182 
51706 
58235 
54768 

71305 
77846 
84391 
90941 
97495 
1-12 04053 
10616 
17183 
23754 
30329 



43493 
50081 
66674 
63271 
69872 
76478 



89702 
96321 
•1302944 
09571 
16203 



50 

51 ! 
52 
53 
54 
55 
56 
57 
58 
59 

60 II 
* I 



29479 
36124 
42773 
49427 
56085 
62747 
69414 
76086 
82761 
89441 
96126 

•1402815 
09508 
16206 
22908 
29615 
.36326 
43041 
49762 
56486 
63215 
69949 
76687 
83429 
90176 
96928 

ID 03684 

41° 



49° 

115 03684 
10445 
17210 
23979 
30754 
37532 
44316 
51104 
57896 
64693 
71495 
78301 
85112 
91927 
98747 

1-16 05571 
12400 
19264 
26073 
32916 
39763 
46615 
53472 
60334 
67200 
74071 
80947 
87827 
94712 

1-17 01601 
08496 
15395 
22298 
29207 
36120 
43038 
49960 
56888 
63820 
70756 
77698 ' 
84644 
91595 
98551 
18 05512 
12477 
19447 
26422 
33402 
40387 
47376 
54370 



68373 
75382 1 
82395 
89414 
96437 
•19 03465 
10498 
17536 
40° I 



50° 


51° 


52° 


53° 


119 17536 


1-23 48972 


1-27 99416 


1-32 70449 


24579 


56319 


1-28 07094 


78483 


31626 


63672 


14776 


86524 


38679 


71030 


22465 


94571 


45736 


78393 


30160 


1-3302624 


52799 


85762 


37860 


10684 


59866 


93136 


45566 


18750 


66938 


1-24 00515 


53277 


26822 


74015 


07900 


60995 


34900 


81097 


15290 


68718 


42984 


88184 


22685 


76447 


51075 


95276 


30086 


84182 


59172 


1-20 02373 


37492 


91922 


67276 


09475 


44903 


99669 


75386 


16581 


52320 


1-29 07421 


83502 


23693 


59742 


15179 


91624 


30810 


67169 


22943 


99753 


37932 


74602 


30713 


1-34 07888 


45058 


82040 


38488 


16029 


52190 


89484 


46270 


24177 


59327 


96933 


54057 


32331 


66468 


1-25 04388 


61850 


40492 


73615 


11848 


69649 


48658 


80767 


19313 


77454 


56832 


87924 


26784 


85265 


65011 


95085 


34260 


93081 


73198 


1-21 02252 


41742 


1-30 00904 


81390 


09424 


49229 


08733 


89589 


16601 


56721 


16567 


97794 


23783 


64219 


24407 


1-3506006 


30970 


71723 


32254 


14224 


38162 


79232 


40106 


22449 


45359 


86747 


47964 


30680 


52562 


94267 


55828 


38918 


59769 


1-26 01792 


63699 


47162 


66982 


09323 


71575 


55413 


74199 


16860 


79457 


63670 


81422 


24402 


87345 


71934 


88650 


31950 


95239 


80204 


95883 


39503 


1-31 03140 


88481 


1*2203121 


47062 


11046 


96764 


10364 


54626 


18958 


1-3605054 


17613 


62196 


26876 


13350 


24866 


69772 


34801 


21653 


32125 


77353 


42731 


29963 


39389 


84940 


50668 


38279 


46658 


92532 


58610 


46602 


53932 


1-27 00130 


66559 


54931 


61211 


07733 


74513 


63267 


68496 


15342 


82474 


71610 


75786 


22957 


90441 


79959 


83081 


30578 


98414 


88315 


90381 


38204 


1-32 06393 


96678 


97687 


45835 


14379 


1-37 05047 


1-2304997 


53473 


22370 


13423 


12313 


61116 


30368 


21806 


19634 


68765 


38371 


30195 


26961 


76419 


46381 


38591 


34292 


84079 


54397 


46994 


41629 


91745 


62420 


55403 


48972 


99416 


70448 


63819 


39° 


38° 


37° 


36° 



54° 

1-37 63819 
72242 

80672 
89108 
97551 

1-3806001 
14458 
22922 
31392 
39869 
48353 
56844 
65342 
73847 
82358 
90876 
99401 

1-3907934 
16473 
25019 

33571 
42131 
50698 
59272 
67852 
76440 
85034 
93636 

1-40 02245 
10860 
19483 
28113 
36749 
45393 
54044 
62702 
71367 
80039 
88718 
97405 

1-41 06098 
14799 
23506 
32221 
40943 
49673 
58409 
67153 
75904 
84662 

93427 
1-42 02200 
10979 
19766 
28561 
37362 
46171 
54988 
63811 
72642 
81480 

35° 



NAT. COTAN. 



NATURAL TANGENTS. 



263 



/ 


65° 


56° 


57° 1 


58° 1 


59° 


60° I 


61° 





1-42 81480 


148 25610 


1-53 9S650 


1-60 03345 1-66 42795 


L-73 20508 


1-80 40478 < 


1 


90326 


34916 


1-54 08460 


13709 


53766 


32149 


52860 


2 


99178 


44231 


18280 


24082 


64748 


43803 


65256 , 


3 


14308039 


53554 


28108 


34465 


75741 


55468 


77664 , 


4 


16906 


62884 


37946 


44858 


86744 


67144 


90086 , 


5 


25781 


72223 


47792 


55260 


97758 


78833 


1-81 U2521 , 


6 


34664 


81570 


57647 


65672 


1-67 08782 


90533 


14969 


7 


43554 


90925 


67510 


76094 


19818 


1-7402245 


27430 


8 


52451 


149 002SS 


77383 


86525 


30864 


13969 


39904 


9 


61356 


09659 


87264 


96966 


41921 


25705 


52391 


LO 


70268 


19039 


97155 


1-61 07417 


52988 


37453 


64892 


LI 


79187 


28426 


1-55 07054 


17878 


64067 


49213 


77405 


L2 


88114 


37822 


16963 


28349 


^5156 


60984 


89932 


L3 


97049 


47225 


26880 


38829 


86256 


72768 


1-82 02473 


L4 


144 05991 


56637 


36806 


49320 


97367 


84564 


15026 


L5 


14940 


66058 


46741 


59820 


1-68 08489 


96371 


27593 


L6 


23897 


75486 


56685 


70330 


19621 


1-75 08191 


40173 


L7 


32862 


84923 


66639 


80850 


30765 


20023 


52767 


L8 


41834 


94367 


76601 


91380 


41919 


31866 


65374 


L9 


50814 


1-50 03821 


86572 


1-62 01920 


53085 


43722 


77994 


20 


59801 


13282 


96552 


12469 


64261 


55590 


90628 


21 


68796 


22751 


1-56 06542 


23029 


75449 


67470 


1-83 03275 


22 


77798 


32229 


16540 


33599 


86647 


79362 


15936 


23 


86808 


41716 


26548 


44178 


97856 


91267 


28610 


24 


95825 


51210 


36564 


54768 


1-69 09077 


1-76 03183 


41297 


25 


145 04850 


60713 


46590 


65368 


20308 


15112 


53999 


26 


13883 


70224 


56625 


75977 


31550 


27053 


66713 


27 


22923 


79743 


66669 


86597 


42804 


39007 


79442 


28 


31971 


89271 


76722 


97227 


54069 


50972 


92184 


29 


41027 


98807 


86784 


1-63 07867 


65344 


62950 


1-8404940 


30 


50090 


1-51 08352 


96856 


18517 


76631 


74940 


17709 


31 


59161 


17905 


1-57 06936 


29177 


87929 


86943 


30492 


32 


68240 


27466 


17026 


39847 


99238 


98958 


43289 


33 


77326 


37036 


27125 


50528 


1-70 10559 


1-77 10985 


56099 


34 


86420 


46614 


37234 


61218 


21890 


23024 


68923 


35 


95522 


56201 


47352 


71919 


33233 


35076 


81761 


36 


14604632 


65796 


57479 


82630 


44587 


47141 


94613 


37 


13749 


75400 


67615 


93351 


55953 


59218 


1-85 07479 


38 


22874 


85012 


77760 


1-64 04082 


67329 


71307 


20358 


39 


32007 


94632 


87915 


14824 


78717 


83409 


33252 


40 


41147 


1-5204261 


98079 


25576 


90116 


95524 


46159 


41 


50296 


13899 


1-58 08253 


36338 


1-71 01527 


1-78 07651 


59080 


42 


59452 


23545 


18436 


47111 


12949 


19790 


72015 


43 


68616 


33200 


28628 


57893 


24382 


31943 


84965 


44 


77788 


42863 


38830 


68687 


35827 


44107 


97928 


45 


86967 


52535 


49041 


79490 


47283 


56285 


1-86 10905 


46 


96155 


62215 


59261 


90304 


58751 


68475 


23896 


47 


147 05350 


71904 


69491 


1-65 01128 


70230 


80678 


36902 


48 


14553 


81602 


79731 


11963 


81720 


92893 


49921 


49 


23764 


91308 


89979 


22808 


93222 


1-79 05121 


62955 


50 


32983 


1-53 01023 


1-59 00238 


33663 


1-7204736 


17362 


76003 


51 


42210 


10746 


10505 


44529 


16261 


29616 


89065 


52 


51445 


20479 


20783 


55405 


27797 


41883 


1-87 02141 


53 


60688 


30219 


3107C 


66292 


39346 


54162 


15231 


54 


69938 


39969 


41366 


77189 


50905 


66454 


28336 


55 


79197 


49727 


51675 


88097 


62477 


78759 


41455 


56 


88463 


59494 


61987 


99016 


74060 


91077 


54588 


57 


9773S 


6927C 


7231S 


\ 1-66 09945 


85654 


1-S0 03408 


67736 


58 


148 07021 


79054 


1- 82641 


20884 


97260 


15751 


80898 


59 


16311 


8884? 


9299] 


L 31834 


1-73 08878 


28108 


94074 


60 


2561C 


9865( 


) 1-60 0334J 


)) 42795 


20508 


40478 


1-88 07265 


/ 


34° 


33° 


32° 


1 31° 30° 


29° 


28° 








: 


&AT. COT 


IN 







264 



NATURAL TANGENTS. 



f 


62° 


63° 


640 


65° 


66° 


67° 


68° 


/ 





1-88 07265 


1-96 26105 


2-05 03038 


2-1445069 


2*24 60368 


2-35 58524 2-47 50869 


60 


1 


20470 


40227 


18185 


61366 


77962 


77590 


71612 


59 


2 


33690 


54364 


33349 


77683 


95580 


96683 


92386 


58 


3 


46924 


68518 


48531 


94021 


2-25 13221 


2-3615801 


2-48 13190 


57 


4 


60172 


82688 


63732 


2-15 10378 


30885 


34946 


34023 


56 


5 


73436 


96874 


78950 


26757 


48572 


54118 


54887 


55 


6 


66713 


1-9711077 


94187 


43156 


66283 


73316 


75781 


54 


7 


L-89 00006 


25296 


2-06 09442 


59575 


84016 


92540 


96706 


63 


8 


13313 


39531 


24716 


76015 


2-26 01773 


2-37 11791 


2-49 17660 


52 


9 


26635 


53782 


40008 


92476 


19554 


31068 


38645 


51 


10 


39971 


68050 


55318 


2-1608958 


37357 


60372 


59661 


50 


11 


53322 


82334 


70646 


25460 


55184 


69703 


80707 


49 


12 


66688 


96635 


85994 


41983 


73035 


89060 


2-50 01784 


48 


13 


80068 


1-98 10952 


2-07 01359 


58527 


90909 


2 38 08444 


22891 


47 


14 


93464 


25286 


16743 


75091 


2-27 08807 


27855 


44029 


46 


15 


1-90 06874 


39636 


32146 


91677 


26729 


47293 


65198 


45 


16 


20299 


54003 


47567 


2-17 08283 


44674 


66758 


86398 


44 


17 


33738 


68387 


63007 


24911 


62643 


86250 


2-51 07629 


43 


18 


47193 


82787 


78465 


41559 


80636 


2-39 05769 


28890 


42 


19 


60663 


97204 


93942 


58229 


98653 


25316 


50183 


41 


20 


74147 


1-99 11637 


2-0809438 


74920 


2-28 16693 


44889 


71507 


40 


21 


87647 


26087 


24953 


91631 


34758 


64490 


92863 


39 


22 


1-91 01162 


40554 


40487 


2-18 08364 


52846 


84118 


2-52 14249 


38 


23 


14691 


55038 


56039 


25119 


70959 


2-40 03774 


35667 j 37 


24 


28236 


69539 


71610 


41894 


89096 


23457 


57117 36 


26 


41795 


84056 


87200 


58691 


2-29 07257 


43168 


78598 ! 35 


26 


55370 


98590 


2-09 02809 


75510 


25442 


62906 


2-53 00111 ! 34 


27 


68960 


2-0013142 


18437 


92349 


43651 


82672 


21655 ' 33 


28 


82565 


27710 


34085 


2-19 09210 


61885 


2-41 02465 


43231 ! 32 


29 


96186 


42295 


49751 


26093 


80143 


22286 


64839 j 31 


30 


1-9209821 


56897 


65436 


42997 


98425 


42136 


86479! 30 


31 


23472 


71516 


81140 


59923 


2-30 16732 


62013 


2-54 08151 29 


32 


37138 


86153 


96864 


76871 


35064 


81918 


29855 ' 28 


33 


50819 


2-01 00806 


210 12607 


93840 


53420 


2-4201851 


51591 | 27 


34 


64516 


15477 


28369 


2-20 10831 


71801 


21812 


73359 ' 26 


35 


78228 


30164 


44150 


27843 


90206 


41801 


95160 j 25 


36 


91956 


44869 


59951 


44878 


2-31 08637 


61819 


2-55169921 24 


37 


1-93 05699 


59592 


75771 


61934 


27092 


81864 


38858 23 


38 


19457 


74331 


91611 


79012 


45571 


2-43 01938 


607561 22 


39 


33231 


89088 


211 07470 


96112 


64076 


22041 


82686 21 


40 


47020 


2-0203862 


23348 


2-21 13234 


82606 


42172 


2-56 04649 


20 


41 


60825 


18654 


39246 


30379 


2-32 01160 


62331 


26645 


19 


42 


74645 


33462 


55164 


47545 


19740 


82519. 


48674 


J8 


43 


88481 


48289 


71101 


64733 


38345 


2-44 02736 


70735 


17 


44 


1-9402333 


63133 


87057 


81944 


56975 


22982 


92830 


16 


45 


16200 


77994 


2-12 03034 99177 


75630 


43256 


2-57 14957 j 15 


46 


30083 


92873 


19030 2-22 16432 


94311 


63559 


37118 ' 14 


47 


43981 


20307769 


35046 33709 


2-33 13017 


83891 


59312! 13 


48 


57896 


22683 


51082 51009 


31748 


2 45 04252 


81539 12 


49 


71826 


37615 


67137 68331 


50505 


24642 


2-58 03S00 11 


50 


85772 


52565 


83213 1 85676 


69287 


45061 


26094' 10 


61 


9973S 


67532 


99308 2-23 03043 


88095 


65510 


48421 9 


62 


1-9513711 


82517 


2-13 15423, 20433 


2-34 06928 


85987 


70782 8 


53 


27704 


I 9751S 


31559, 37845 


25787 


2.46 06494 


9P177 1 7 


54 


4171c 


5 2*0412540 


47714' 55280 


44672 


27030 


2-59 15606 6 


55 


55735 


) 27578 


63890, 72738 


63582 


47596 


38068 5 


56 


6978( 


) 42634 


80085 90218 


82519 


68191 


60564 4 


57 


8883' 


57708 


96301 2-24 07721 


2-35 01481 


88816 


83095 3 


58 


979K 


) 7280C 


) 2-14 125371 25247 


20469 


2-47 09470 


2'60 05659 2 


59 


1-96 1200< 


) 879K 


) 28793' 42796 


39483 


30155 


28258 1 1 


eo 


2610 


i 2-05 0303* 


* 45069, 60368 


58524 


50869 


508911 


/ 


27° 


26° 


25° 1 24° 


23° 


22° 


21° | ' 










NAT. COT 


iN. 









NATURAL TANGENTS. 



265 



69° 

2-60 50891 
73558 
96259 

2-61 18995 
41766 
64571 
87411 

2-62 10286 
33196 
56141 
79121 

2-63 02136 
25186 
48271 
71392 
94549 

2-6417741 
40969 
64232 
8753] 

2-65 10867 
34238 
57645 
81089 

2-66 04569 
28085 
51638 
75227 
98853 

2-67 22516 

46215 
69951 
93725 

2-68 17535 
41383 
65267 
89190 

2-69 13149 
37147 
61181 
85254 

2-70 09364 
33513 
57699 
81923 

2-71 06186 
30487 
54826 
79204 

2-72 03620 

28076 
52569 
77102 

273 01674 
26284 
50934 
75623 

2-74 00352 
25120 
49927 
74774 

20° 



70 c 

1-74 74774 
99661 

•75 24588 
49554 
74561 



1-76 24695 

49822 

74990 

1-77 00199 

25448 

50738 

76069 

1-78 01440 

26853 

52307 

77802 

;-79 03339 

2891' 

5453' 

80198 

2-80 05901 
31646 
57433 
83263 

2'81 09134 
35048 
61004 
87003 

2-82 13045 
39129 
65256 
91426 

2-83 17639 
43896 
70196 
96539 

2-84 22926 
49356 
75831 

2-85 02349 
28911 
5551' 
82168 

2-86 08863 
35602 
62386 
89215 

2-87 16088 
4300' 
69970 
96979 

2-88 24033 
51132 
78277 

2-89 05467 
32704 
59986 
87314 

2-90 14688 
42109 

19° 



71 

2-90 42109 
69576 
97089 

2-91 24649 
52256 
79909 

2-9207610 
35358 
63152 
90995 

2-9318885 
46822 
74807 

2-94 02840 
30921 
59050 
87227 

2-95 15453 
43727 
72050 

2-96 00422 
28842 
57312 
85831 

2-97 14399 
43016 
71683 

2-98 00400 
2916: 
5798! 
86850 

2-99 15766 
44734 
73751 

3-00 02820 
31939 
61109 
90330 

3-01 19603 
48926 

78301 

3-02 07728 
37207 
66737 
96320 

3-03 25954 
55641 
85381 

30415173 
45018 
74915 

3-05 04866 
34870 
64928 
95038 

3-06 25203 
55421 
85694 

3-07 16020 
46400 
76835 

18° 



72° 

•07 76835 

■•08 07325 
37869 
68468 
99122 

•09 29831 
60596 
91416 

•10 22291 
53223 
84210 

•11 15254 
46353 
77509 

•12 08722 
39991 
71317 

•13 02701 
34141 



97194 

3-14 28807 
60478 
92207 

3-15 23994 
55840 
87744 

31619706 
51728 
83808 

3-17 15948 
48147 
80406 

3-18 12724 
45102 
77540 

3-1910039 
42598 
75217 

3-20 07897 
40638 
73440 

3-21 06304 
39228 
72215 1 

3-22 05263 
38373 
71546 

3-23 04780, 
38078 
71438 

3-2404860? 
38346 1 
71895 i 

3-25 05508 
39184 | 
72924 J 

3-26 067281 
40596' 
74529 c 

3-27 08526| 

17° 



73° 


74° 


75° 


3-27 08526 


3-48 74144 


3-7320508 


42588 


3-49 12470 


63980 


76715 


50874 


3-7407546 


3-28 10907 


89356 


51207 


45164 


3-50 27916 


94963 


79487 
3-29 lc&76 


66555 


3-75 38815 


3-51 05273 


82763 


48330 


44070 


3-76 26807 


82851 


82946 


70947 


3-30 17438 


3-52 21902 


3-77 15185 


52091 


60938 


59519 


86811 


3-5300054 


3-78 03951 


3-31 21598 


39251 


48481 


56452 


78528 


93109 


91373 


3-54 17886 


3-79 37835 


3-32 26362 


57325 


82661 


61419 


96846 


3-80 27585 


96543 


3-55 36449 


72609 


3-3331736 


76133 


3-81 17733 


66997 


3-56 15900 


62957 


3-3402326 


55749 


3-82 08281 


37724 


95681 


53707 


73191 


3-57 35696 


99233 


3-35 08728 


75794 


3-8344861 


44333 


3-58 15975 


90591 


80008 


56241 


3-84 36424 


3-36 15753 


96590 


82358 


51568 


3-5937024 


3-85 28396 


87453 


77543 


74537 


3-37 23408 


3-60 18146 


3-8620782 


59434 


58835 


67131 


95531 


99609 


3-87 13584 


3-38 31699 


3-61 40469 


60142 


67938 


81415 


3-88 06805 


3-39 04249 


3-62 22447 


53574 


40631 


63566 


3-89 00448 


77085 


3-6304771 


47429 


3-40 13612 


46064 


94516 


50210 


87444 


3-9041710 


86882 


3-6428911 


89011 


3-41 23626 


70467 


3-9136420 


60443 


3-65 12111 


83937 


97333 


53844 


3-9231563 


3-42 34297 


95665 


79297 


71334 


3-6637575 


3-93 27141 


3-43 08446 


79575 


75094 


45631 


3-67 21665 


3-94 23157 


82891 


63845 


71331 


3-4420226 


3-68 06115 


3-95 19615 


57635 


48475 


68011 


95120 


90927 


3-9616518 


3-45 32679 


3-69 33469 


65137 


70315 


76104 


3-97 13868 


3-4608026 


3-70 18830 


62712 


45813 


61648 


3-98 11669 


83676 


3-71 04558 


60739 


3-47 21616 


47561 


3-99 09924 


59632 


90658 


59223 


97726 


3-72 33847 


4-00 08636 


3-48 35896 


77131 


58165 


74144 


3-73 20508 


4-01 07809 


16° 


15° 


14° 



NAT. COTAN. 



266 



NATURAL TANGENTS. 



76° 


77° 


78° 


79° 


80° 1 


81° 


82° | 


/ 


4-01 07809 


4-3314759 < 


1*70 46301 


3-1 445540 


5-6712818 1 


3 3137515 71153697J 60 


57570 


72316 


471 13686 


525557 


809446 


256601 


304190 59 


4-02 07446 


4-34 30018 


81256 


605813 


906394 


376126 


455308 58 


57440 


87866 


4-72 49012 


686311 


5-7 003663 


496092 


607056 57 


4-03 07550 


4-35 45861 


4-73 16954 


767051 


101256 


616502 


759437, 56 


57779 


4-36 04003 


85083 


848035 


199173 


737359 


912456 55 


4-0408125 


62293 


4-7453401 


929264 


297416 


858665 


7-2 060116 54 


58590 


4-37 20731 


4-75 21907 


3-2 010738 


395988 


980422 


220422 53 


4-05 09174 


79317 


90603 


092459 


494889 


6-4102633 


375378 52 


59877 


4-38 38054 


4-76 59490 


174428 


594122 


225301 


530987 


51 


4-06 10700 


96940 


4-77 28568 


256647 


693688 


348428 


687255 


50 


61643 


4-39 55977 


97837 


339116 


793588 


472017 


844184 


49 


4-07 12707 


4-40 15164 


4-78 67300 


421836 


893825 


596070 


7-3 001780 


48 


63892 


74504 


4-79 36957 


504809 


994400 


720591 


160047 


47 


4-08 15199 


4-41 33996 


4-80 06808 


588035 


5-8 095315 


845581 


318989 


46 


66627 


93641 


76854 


671517 


196572 


971043 


478610 


45 


4-09 18178 


4-42 53439 


4-8147096 


755255 


298172 


6-5 096981 


638916 


44 


69852 


4-43 13392 


4-82 17536 


839251 


400117 


223396 


799909 


43 


4-10 21649 


73500 


88174 


923505 


502410 


350293 


961595 


42 


73569 


4-44 33762 


4-83 59010 


5-3 008018 


605051 


477672 


7*4123978 


41 


4-1125614 


94181 


4-8430045 


092793 


708042 


605538 


287064 


40 


77784 


4-45 54756 


4-85 01282 


177830 


811386 


733892 


450855 


39 


4-12 30079 


4-4615489 


72719 


263131 


915084 


862739 


615357 


38 


82499 


76379 


4-86 44359 


348696 


5-9 019138 


992080 


780576 


37 


4-13 35046 


4-47 37428 


4-87 16201 


434527 


123550 


6-6 121919 


946514 


36 


87719 


98636 


88248 


520626 


228322 


252258 


7-5 113178 


35 


4-1440519 


4-48 60004 


4-88 60499 


606993 


333455 


383100 


280571 


34 


93446 


4-49 21532 


4-89 32956 


693630 


438952 


514449 


448699 


33 


4-15 46501 


83221 


4-90 05620 


780538 


544815 


646307 


617567 


32 


99685 


4-50 45072 


78491 


867718 


651045 


778677 


787179 


31 


4-16 52998 


4-51 07085 


4-91 51570 


955172 


757644 


911562 


957541 


30 


4-17 06440 


69261 


4-92 24859 


5-4 042901 


864614 


6-7 044966 


7-6 128657 


29 


60011 


4-5231601 


98358 


130906 


971957 


178891 


300533 


28 


4-1813713 


94105 


4-93 72068 


219188 


6-0 079676 


313341 


473174 


27 


67546 


4-53 56773 


4-94 45990 


307750 


187772 


448318 


646584 


26 


4-19 21510 


4-54 19608 


4-95 20125 


396592 


296247 


583826 


820769 


25 


75606 


82608 


94474 


485715 


405103 


719867 


995735 


24 


4-20 29835 


4-55 45776 


4-96 69037 


575121 


514343 


856446 


7-7 171486 


23 


84196 


4-56 09111 


4-97 43817 


664812 


623967 


993565 


348028 


22 


4-21 38690 


72615 


4-98 18813 


754788 


733979 


6-8 131227 


525366 


21 


93318 


4-57 36287 


94027 


845052 


844381 


269437 


703506 


20 


4-2248080 


4-58 00129 


4-99 69459 


935604 


955174 


408196 


882453 


19 


4-2302977 


64141 


5-0045111 


5-5 026446 


6-1 066360 


547508 


7-8 062212 


18 


58009 


4-59 28325 


5-01 20984 


117579 


177943 


687378 


( 242790 


17 


4-24 13177 


92680 


97078 


209005 


289923 


827807 


424191 


16 


68482 


4-60 57207 


5-0273395 


300724 


402303 


968799 


606423 


15 


4-25 23923 


4-61 21908 


5-0349935 


392740 


515085 


6-9 110359 


789489 


14 


79501 


86783 


5-04 26700 


485052 


628272 


252489 


973396 


13 


4-26 35218 


4-62 51832 


5-05 03690 


577663 


741865 


395192 


7-9 158151 


12 


91072 


4-63 17056 


80907 


670574 


855867 


538473 


343758 


11 


4-27 47066 


82457 


5-06 58352 


763786 


970279 


682335 


530224 


10 


4-28 03199 


4-6448034 


5-07 36025 


857302 


6-2085106 


826781 


717555 


9 


59472 


4-65 13788 


5-08 13928 


951121 


200347 


971806 


905756 


8 


4-29 15885 


79721 


92061 


5-6045247 


316007 


7-0 117441 


8-0 094835 


7 


72440 


4-66 45832 


5-09 70426 


139680 


432086 


263662 


284796 


e 


4-30 29136 


4-67 12124 


5-10 49024 


234421 


548588 


410482 


475647 


5 


85974 


78595 


5-11 27855 


329474 


665515 


557905 


667394 


4 


4-31 42955 


4-68 45248 


5-12 06921 


424838 


782868 


705934 


860042 


3 


4-32 00079 


4-69 12083 


86224 


520516 


900651 


854573 


8-1 053599 


2 


57347 


79100 


5-13 65763 


616509 


6-3 018866 


7-1 003826 


248071 


1 


4-3314759 


4-70 46301 


5-14 45540 


712818 


137515 


153697 


443464 





13° 


12° 


11° 

Hi 


10° 

LT, COTA, 


9° 


8° 


7° 


/ 



NATURAL TANGENTS. 



267 



83° 


84° | 


85° 


86° 


87° I 


88° 


89° 


/ 


81443464 


9-5 143645 


11.430052 


14-300666 


19-081137 


28-636253 


57-289962 


60 


639786 


410613 


468474 


360696 


187930 


877089 


58-261174 


59 


837041 


679068 


507154 


421230 


295922 


29-122006 


59-265872 


58 


8-2 035239 


949022 


546093 


482273 


405133 


371106 


60-305820 


57 


234384 


9-6 220486 


585294 


543833 


515584 


624499 


61-382905 


56 


434485 


493475 


624761 


605916 


627296 


882299 


62-499154 56 


635547 


768000 


664495 


668529 


740291 


30-144619 


63-656741 


54 


837579 


9-7 044075 


704500 


731679 


854591 


411580 


64-858008 


53 


8-3 040586 


321713 


744779 


795372 


970219 


683307 


66-105473 


52 


244577 


600927 


785333 


859616 


20-087199 


959928 


67-401854 


51 


449558 


881732 


826167 


924417 


205553 


31-241577 


68-750087 


50 


655536 


9-8 164140 


867282 


989784 


325308 


528392 


70-153346 


49 


862519 


448166 


908682 


15-055723 


446486 


820516 


71-615070 


48 


8-4 070515 


733823 


950370 


122242 


569115 


32-118099 


73-138991 


47 


279531 


9-9 021125 


992349 


189349 


693220 


421295 


74-729165 


46 


489573 


310088 


12.034622 


257052 


818828 


730265 


76-390009 


45 


700651 


600724 


077192 


325358 


945966 


33-045173 


78-126342 


44 


912772 


893050 


120062 


394276 


21-074664 


366194 


79-943430 


43 


8-5 125943 


10-018708 


163236 


463814 


204949 


693509 


81-847041 


42 


340172 


048283 


206716 


533981 


336851 


34-027303 


83-843507 


41 


555468 


078031 


250505 


604784 


470401 


367771 


85-939791 


40 


771838 


107954 


294609 


676233 


605630 


715115 


88-143572 


39 


989290 


138054 


339028 


748337 


742569 


35-069546 


90-463336 


38 


8-6 207833 


168332 


383768 


821105 


881251 


431282 


92-908487 


37 


427475 


198789 


428831 


894545 


22-021710 


800553 


95-489475 


36 


648223 


229428 


474221 


968667 


163980 


36-177596 


98-217943 


35 


870088 


260249 


519942 


16-043482 


308097 


562659 


101-10690 


34 


8-7 093077 


291255 


565997 


118998 


454096 


956001 


104-17094 


33 


317198 


322447 


612390 


195225 


602015 


37-357892 


107-42648 


32 


542461 


353827 


659125 


272174 


751892 


768613 


110-89205 


31 


768874 


385397 


706205 


349855 


903766 


38-188459 


114-58865 


30 


996446 


417158 


753634 


428279 


23-057677 


617738 


118-54018 


29 


8*8 225186 


449112 


801417 


507456 


213666 


39-056771 


122-77396 


28 


455103 


481261 


849557 


587396 


371777 


505895 


127-32134 


27 


686206 


513607 


898058 


668112 


532052 


965460 


132-21851 


2C 


918505 


546151 


946924 


749614 


694537 


40-435837 


137-50745 


25 


8-9 152009 


578895 


996160 


831915 


859277 


917412 


143-23712 


24 


386726 


611841 


13-045769 


915025 


24-026320 


41-410588 


149-46502 


22 


622668 


644992 


095757 


998957 


195714 


915790 


156-25908 


22 


859843 


678348 


146127 


17-083724 


367509 


42-433464 


163-70019 


21 


9-0 098261 


711913 


196883 


169337 


541758 


964077 


171-88540 


2( 


337933 


745687 


248031 


255809 


718512 


43 508122 


180-93220 


IS 


578867 


779673 


299574 


343155 


897826 


44-066113 


190-98419 


M 


821074 


813872 


351518 


431385 


25-079757 


638596 


202-21875 


17 


9-1 064564 


848288 


403867 


520516 


264361 


45-226141 


214-85762 


1C 


309348 


882921 


456625 


610559 


451700 


829351 


229-18166 


U 


555436 


917775 


509799 


701529 


641832 


46-448862 


245-55198 


14 


802838 


952850 


563391 


793442 


834823 


47-085343 


264-44080 


11 


9-2 051564 


988150 


617409 


886310 


26-030736 


739501 


286-47773 


IS 


301627 


11-023676 


671856 


980150 


229638 


48-412084 


312-52137 


11 


553035 


059431 


726738 


18-074977 


431600 


49-103881 


343-77371 


1C 


805802 


095416 


782060 


170807 


636690 


815726 


381-97099 


£ 


9-3 059936 


131635 


837827 


267654 


844984 


50-548506 


429-71757 


8 


315450 


168089 


894045 


365537 


27-056557 


51-303157 


491-10600 


7 


572355 


204780 


950719 


464471 


271486 


52-080673 


572-95721 


e 


830663 


241712 


14-007856 


564473 


489853 


882109 


687-54887 


s 


9-4 090384 


278885 


065459 


665562 


711740 


53-708587 


859-43630 


4 


351531 


316304 


123536 


767754 


937233 


54-561300 


1145-9153 


a 


614116 


353970 


182092 


871068 


28-166422 


55-441517 


1718-8732 


2 


878149 


391885 


241134 


975523 


399397 


56-350590 


3437-7467 


1 


9.5 143645 


430052 


300666 


19-081137 


636253 


57-289962 


Infinite. 





6° 


5° 


4° 


3° 

AT. COTi 


2° 

jr. 


1° 


0° 


/ 



268 



SLOPES, FOR TOPOGRAPHY. 



TABLE XV. 

SLOPES, FOR TOPOGRAPHY. 





Degrees. 


Vertical Rise 

in ioo 
Horizontal. 


Horizontal 

Distance 

to a Rise of 

10. 


Degrees. 


Vertical Rise 

in 100 
Horizontal. 


Horizontal 

Distance 

to a Rise of 

10. 


X 


* »75 


572.9 


19 


34-43 


29.0 


2 


3-49 


286.4 


20 


36.40 


27-5 


3 


5-24 


190.8 


21 


38.40 


26.0 


4 


6.99 


143.0 


22 


40.40 


24.7 


5 


8.75 


114. 3 


23 


42.45 


23-5 


6 


10.51 


95-i 


24 


44-52 


22.4 


7 


12.28 


81.4 


25 


46.63 


21.4 


8 


14.05 


71.2 


26 


48.77 


20.5 


9 


15.83 


63.1 


27 


5o-95 


19.6 


IO 


17-63 


56.7 


28 


53.17 


18.8 


ii 


19.44 


5i.4 


29 


55-43 


18.0 


12 


21.25 


47.0 


30 


57-73 


17-3 


13 


23.09 


43-3 


35 


70.02 


14.2 


14 


24-93 


40.1 


40 


83.91 


xi. 9 


15 


26.79 


37-3 


45 


100.00 


IO. Q 


16 


28.67 


34-9 


50 


119. 17 


8.4 


»7 


30-57 


32.7 


55 


142.81 


7.O 


18 


32.49 


30-7 


60 


173- 20 


5-7 



Note. — See page 52, Art. XVIII., for examples in the use of 
Table XV. 




TABLE XVI. 

CHORDS, MIDDLE ORDINATES, EXTERNAL SE- 
CANTS, AND APEX DISTANCES OF A ONE- 
DEGREE CURVE. 

The angles of the table are the 
intersection angles, I, equal to 
the total central angle included 
between the tangent points. 

To find the corresponding func- 
tion for any other curve, divide 
the tabular number by the de- 
gree of curvature. 

The unit chord is assumed to 
be one hundred feet long. 

By using radius of 5,730 feet, 
the chord column of the table can be made serviceable for 
plotting. 

To use the table for curves having chords of 20 metres each, 
divide the several tabular functions, that is to say, the Chord, 
Ordinate, Ex-sec, and Apex Distance, by 5 times the proposed 
metric degree of curve, for the proper values of said functions 
in metric measure. Thus, for a 2° metric curve, chords 20 
metres each, the divisor would be 2 X 5 = 10; for a 2° 30' 
curve, 12.5, and so on; taking care to reduce minutes to deci- 
mals of a degree before making the multiplication for a divisor 
in each case. 

Also, if the length in metres of any proposed Chord, Ordi- 
nate, Ex-sec or A. D. for a given angle be known, the tabular 
function corresponding to that angle divided by 5 times the 
known chord, ordinate, etc., will ascertain the degree of 
curvature for chords of 20 metres each, using the foregoing 
precaution as to decimals. 

Example 1.-4° curve, intersection angle 48°, 20-metre 
chords, . \ apex dist. = 2551.1 -f- 5 X 4 = 127.55 metres. 

Example 2. — Apex distance 200 metres, angle 58° 20', .*. de- 
gree of metric curve = 3198 + 200 X 5 = 3 T \ degrees = 3° IV > 
chords being 20 metres each. 



270 



FUNCTIONS OF A ONE-DEGREE CURVE. 



55 


« *vO CO « -*vO WON M-vO WON -*-vO CO N «fvO 00 N "*-vO 00 
H w h m m cm cm cm cm cm rorororocO'*-'«-Tj-'*t--«-inin m\r> invo 




O NfiO tsfoO NfOO t% ro smo smo t»» ro o t>. ro o smo ncoO 
O vo 000*0 MO^O fOOvO ro O vo roovO ro vo MO^O ro O vo roOvO ro 

m ro invo oo O m co mvo oo O h ro iovo co m ro mvo oo m ro mvo oo 6 
mmmmio mvo vo vo vo <o vo t^t^t^.t>.t^ t-^oo oooooooooo o^CM^aa^o 



CO rOOO rj- t>» •"*■ CM OCOCO t** txOO O H COVO Ov CM vO O m m vo rooco in^fO 
m ro -<*-vo co Ov m ro invo co O CM t*-vo O h ro ioco ro inco o ro>o co w ■<*■ t> 

04 CM CM CM CM CM COrOrOrOrO'^-'^-^'^-ThlOlOlO lOvO vO^O'O t^ t-» t-» C^.00 CO CO 

0060000000006006060666066606606 



00 roco -*• t^.-«j-CM O0000 t^ t>.0O Ov h rovo Ov CM vo in m vo rooco in-tro 
h ro Tt-vO CO Ov m ro invo CO CM tJ-vo Ov m ro 100O ro iOCO rovo CO m -4- t^ 
CM CM CM CM CM CM rOfOfOrOCO'"4--^--^Tl--^lOlO»0 lOVO ^HOvO t>» t^ tv. tvCO CO 00 

6666666666666666666666666666606 



Q 


8 


fOvO 


8 


rovO 


8 


rovo 


O 


rovo 


O 


rovO 


8 


rovo 





rovo 





rovo 





rovo 


O 


rovo 00 


K 


rovo 


rovo 


rovo 





rovo 


O 


rovo 


rovo 


O 


rovo 





rovo 





rovo 


O 


rovo 


Ov 


O 


8 


rovo 





rovo 





rovo 


O 


rovo 


O 


rovo 





rovo 





rovo 





rovo 





rovo 


& 


rovo 


0> 




O 






CM 


CM CM 


ro ro ro 




«*• -<*■ 


in 10 mvo vo vo 




t^ t^.00 CO CO 





w N "^-VO Ov CM lOON^OviOHOO tJ-CM OvOO t>* C^ 1^.00 m tj-vO O -<J-CO 

SyQQQOOMMMCMCMroTj-^ mvo r^ t^co ov o h cm ■«*■ mvo soo h 
OOOOOOOOOOOOOOOOOOOOOhhhhhmmhcmcm 

6666666666666666666666666666666 



O Q M CM tJ-vO O CM in f> ■* CMO H 00 iJ-CM o OVOO t^ t>s f»»00 H tJ-vO '^■00 
QOOOOOOMHHCMCMrO'*"'*- mvo t^ t>.oo Ov m cm ■<*• invo NO^O m 
0°000°0°°°0000^000000hmmhhmhm<>j<N 

6666666666666666666666666666666 



CM ""♦"V© 00 N -^-VO 00 CM -xt-vO 00 CM -^-vo 00 CM -<*-vO 00 CM "^vO 00 Q 

h m h h m cm cm cm cm cm f<irorororO'<-'*-*'**«oiniom mvo 



FUNCTIONS OF A ONE DEGREE CURVE. 



27\ 



tJ- On -<*• Q\ -<f OnvO rO Nlfl'+romfOMlONOH tJ-00 CO t^ CM tv^HOO m CM 

vo o m o> Thoo moo mt^csi t-- cm t^ cm r-* cm nmoo rooo ■<*• on m o *o cm t-^ ro on 
ONOOOwHCMCs , roro-<i-'«**m m>o vO t^ t-^oo oo o^oo O h n « nto^t 

HN«NNCItJ«ciciticiN«CIN«N«NN«flOrtrOtOmfOt<)M 



tJ-00 rooo rooo in CM OnvO -tfOfON CM CM -«J-vO 00 fONN^O MVO rO t^rJ-H 

vo o m o> -<+-oo rooo cm r^N r».<N t^(N t^ cm tswoo rooo tj- on m o no cm sroo 
a>OOOHHNMroro-^-"^-io mvO vO t^ t-»0O ooONONOOnCMCMrorOTt-o*- 

MNNNNNNNWNNNNNNNNNNNNNMroMrOrOfOrOfOCO 



VO ON m On CM moo CM moo h inOO m TJ-t^O rJ-t^O r}-t^0 'tNO -* t^ o •* 
On CM no On CM <0 On CM vO On CM vO On CM no On CM no OWvO On CM vO On CM vO On CM vO On 

On rovO On rovO On rOO On rovO . On conO On rovO On rOvO On rovO On cOvO On rovo On 
OnOO O m m m CM CM cm mrnrrt'tTl-ioiri invO vOvO NS t^OO OO OO On On On 
cm rocorocororororororororororororO(Orororororororororororororo 



} m invo vo nn r^oo oo on 0> 



rON N N M "<*-VO 00 H ^00 CM <0 N00 ■<(■ O Mfl M N CM N CM CM CM ^^OCO m ^> 

t^ rovo On cm moo cm moo cm in On cm no r^ro nh in on ro t-^ m in o« ro t^ cm vo 

00 OnOnOnOnO O M H h CM CM CM 00 CO CO ■*- -<i- in in mvO vO NN t^OO OO On O* 



o o o i 



On cOnO On rOvO On rONO On rovO On rOMD On rovO On 0OnO On rOMD On rovO On ro<0 On 
0\OOOHHMN«Nmmmt't't-inin iomd vo^o sn r-~oo oo oo on on on 

MCTCMCMCMCMCMCMCMCMCMCMNMWCMCMCMNCMCMCMCMCMCMCMCMCMCMCMCM 



272 



FUNCTIONS OF A ONE-DECREE CURVE. 





Hi 


55 


« ^-VO CO N -«-vO 00. O N "^VO CO N ^vO 00 O N ^vO CO N "»-vO 00 

m m m h m n n N N n fOcorofOfO-^-^-'^-'^-'*-in»oiftin iovo 




t*» •>* m CO ion OvO + S^hoo tj-hoo io co s*h owo mo t^ io n 
hoo iomoo inooo io n co io n co io«oo »o n omoh on io n onvo n onvo co 

m coinvOOO m co iovo co h miovooo O h roiovooo m co iovo co O 
lOlfllOlfllfl IOVO vOvOvO^OvO t^.t^t^t^t^ t>.co oooooooooo o>o\o^(M>ao 
C*0*N«NW.NNNNNNNNNNNNNNNNNNNNNNNNCO 


w 

CO 


OiMNH inOilflHOO IO N OnCO «^VO t^00 ON H COvO O -^-00 N SrtOMOH 

io co o co io n ooo mtOM owo -*w o oo vo *cih onco vo -*- co m o co r>.vo 
•«f iovo <0 t->00 O>^0 m « « (O* mvO vo t^CO ON O O m N co ■<*■ iovo vO C^oO 


inminiommin »ovo vovovovovovovovovovovo nnnnnsnnnns 


Q 

o 


<<t- 1-* m \no\fOo>>n« onvo -<*-nmoonOhnco iooo h ino\ cooo coco •«*■ o 

ION O tx ■«*• N ON t^ IO N OOOVO "<f N ONCO VO "<*• N OOO MO fO M O On t^.vO IO 

m in io m ui m io iovo vovovovovovovovovovovo nnnnnnnnnnn 


Q 
K 
O 
M 
U 


Wh *N0 COvO On N lOCO H «*• t> CO t^ COvO On N 1O00 w •«- f^ COvO 
00 N 1O00 N IOOO M IOOO H moo H IOOO H lOOO M Tt-00 M -«-00 M tJ-OO M -ot-OO 

On COvO ON COvO On COvO On covO On COvO On COvO On COvO On COvO On covO On COvO ON 
0>OOOHHM«N«Mron^*r)-inm iovo OvO NS t^CO oo 00 On On On 
'tioioinioiniommioioinioiominioinioioinmioinioioiflinioinio I 


2 


« *4"VO00 N "*NO 00 N <*-vO 00 N "^-VO 00 N "fvO CO N rJ-vO 00 

h m m h m n n N N n forofororo'ot-'oj-'^'oj-'oi-inioioio iovo 


• * 




2 


*» "*vO CO N "<fvO CO N <<fvO CO N ^VO CO N <*vO 00 N -^vO 00 Q 
m m m m m m w « N « fororororO'^'^'^f'oj-ioiou-jin iovO 


< 


OnO COO t^"ol-M SiriMOO ION ONVO fOO ts. ■»*- H 00 1ON00 vy»N ONVO CO S 

O t^-«J-t-t t^-ot-M t^-*»-M t>»-*J-H StJ-MOO TfMOO TfMOO *H00 -ol-HOO IflH 

m ro mvo oo O h m mvo oo m co iovo co O h co mvo oo m co iovO oo 

O00000MMMHHMNCJNNWNCOCOCOCOCOCO'«*-'0hTfTj-TJ--«l-lO 
««NWNW«N«NNWNN«WNNNNN«WNNN«WWNN 


o 

w 

CO 


N M H COlO^ONCOt^HNO H N ■+ M 00 UN * rt N M H CO WN OJ 
ONIOM t^COONlOM t^CO ONVO N ONIONCO lO(NCO ION ONNO CO O t^ -^ M 00 lO 
-ot- IOVO VO t-% t^OO U0»0 O H N« n + r)- IOVO NO t^OO 00 ao M H N wro* 

cococococococoMco4-4-4-4-^4-'*^^^4-4-'*'oj-'ot.in»o»oio»ovo»o 


Q 

as 

O 

Q 


m ONOO tN. (VOO N -ohvO •ot-oo cooo ^-0 r* ■*• H OnOO t^NO t^co n 3- 
ONiOOvO NOO -*m t*»cO ONVO NOO 1OH00 »OhCO lONOO m Cj OnvO CO m CO 10 
* »OvO vO ts, txoo OnOnOOmNNCO^-* iovO vO t^OO oOOnOOmNcocO'* 

cococococococococo4-4-4-^4'4 , ^^^^^^^^'* ,r > l ^ l ^ ir > ,r > ,r)irs 


d 

OS 

o 

X 


tNO COvO On N vO ON N lOCO h -ohOO h t)-NO covO On COVO On N lOCO w ^00 
OvNvO ONN IOONN IOONN lOONN lOONN XOONN »OCO N lOOO N lOCO N lOCO 


On COvO On COvO On COVO On COvO On COvO On COvO On COvO On COvO On covO On covO On 
ONOOOMMMNNNCOCOCOTf-»t--«*-lOin ITJVO OVO NN t^CO 00 CO Ov ON ON 


55 

2 


« '♦■vOOO N "ftWOCO N "1-vOOO N fvO 00 N "^vO 00 N "«*-vO 00 Q 



FUNCTIONS Of A ONE-DEGREE CURVE. 



273 









CM "^-vO 00 CM -*vO 00 CM "*vO 00 CM tJ-vO 00 O CM "*-vO 00 CM "*vO 00 O 

h h m m h cm cm cm cm cm rornrncoro't^ + 't^inifiioin mvo 


Q 

< 


tJ- h OnvO CO hi 00 lOfOO MAN OvO ■"*• CM ON^N O MAN 000 lfl«H 
■^-m r^s ->a- m oo tHM mHOO iohoo mooo ms onnO cm o>no co onnO roO N 

O cm co m t^*oo O n mio t-^oo o n mm r^oo w rom t*»oo cm co io t^ o 
IT) u"> lO IT) ID i/}vO OOvO^OO t^t^t^t^.t^ r^oo oooooooooo OM^O*OM^O>0 


u 

w 

C/3 

X 


m m CM CM MfOfO^-'+m mvO t^OO On i- 1 CM co ■<*■ mvO 00 On h CM co mvO 00 on 
t^«oo on O m cm m •<*■ io\o c^oo o« m co ■*- mvO t^oo 00 h n* iono t^oo on 

OOOhimmmmmmmmhiCMCMCMCMcmcmcmcmcmcococococococococO 


Q 

C 

c 

5 


OnOnOOhmmCMCMCOCO'* iovO t^ t^00 On O m CM CO ■*• lOvO f^00 0> M NO 
O NOO m N fO>t iOnO C^OO ON O M CM CO -*-nO t^.00 On m CM CO t»- in t^00 ON 


p 

o 

ac 
u 


O covo 00 m tj-vO On m co*0 On m -t S o> N in N CO iO00 w covO On t-t * t*» 
NO 0> CM ir> On CM U1O0 CM UIOO M 1/100 M tJ-OO M tJ-00 m ^-t^M "<*- t^ O * t^ "<*■ 


On CM NO On CM vO On CM vO On CM vO On CM nO On CM NO On CM no On CM nO On CM vO On CM vO 0\ 
ONOOO>-i>-iHiCMCMCMrococOTt--<i-"^-ir>u - N ionO vOvO NN t^00 00 00 ON On On 


fc 

s 


CM TfvO 00 CM "^-vO MOM -<J-vO 00 CM -^J-nO 00 CM "«-vO 00 N ^vO 00 
h m m m m cm cm cm cm cm roMmnro^'t^^'tioioinin mvo 


£ 


S5 


CM "*NO 00 CM tj-nO 00 O CM *vO 00 O CM '"J-vO 00 N i*-vO 00 CM -*vO 00 
m t-i M Hi M CM CM CM CM CM fOMrOMfO't^^'t'tlfllfllftlfl lT>vO 


Q 
< 


N-tHOO lO CM OnnO COO N'+HOO U"> CM ON S ■+ m 00 IOCO0 t^^HiOOvO , ^- 
co OnvO co OnvO co OnnO coOnO coO^O coOvO coO NtOO t^^-0 C>. ^- t^ ■*■ 


Q M CO lONO 00 O HI CO U-) C^OO O CM CO lO t^OO CM CO ID 1^00 CM CO lO t^OO O 

OOOOOOMMMHMMNNN«nNmcomfon(Oi-*t***m 
cococococococococococococococococococococococococococococococo 


u 

w 

X* 


00 0\0 h N mtt IOVO r^OO On O h CM CO ■* VOO 1^00 On m CM CO -if lOvO (^ 


t^ t^OO O00OO000000000000000 OnOnOnOnOnOnOnOnOnOnO 


Q 



Q 

5 


00000000000"*On cooo cm nhvO Q m^^co 

iO "4- co CM m OnOO t^vQ io-^-cococm CM hi m O O OnOnOnOnOnOnOnOnOnOnOn 
00 aO m N com* lONO tvOO On hi CM CO ■«*■ invO nO t*«00 On O hi CM CO •* iT)\Q 


t^ t^OO 0000O0O000000OO00O0O OnOnOnOnOnOnOnOnOnOnOnO 




Q 
K 
O 
£ 

U 


fONO 00 M *vO On CM rh tN. O CM irNOO O COVO ON CM •«*• t^ CM IO00 fOVO 00 
00 hi Tj-t>.Hi rj-r>»0 *t^0 *t^O rotsO covO covO O covO On co<0 On CM no 


On covO On COvO On COVO On COvO On COnO On COnO On covO On COnO On CM NO On CM nO On 
0>OOOH'HiHtCMCMCMcococO"5t-Tj-''*-inin m\o ^ovO ns t^.oo oo oo On On On 
invO vOnOnOnO^OvOnOnOvOnC'NOnOnOvOvOvOnOnOnOnOnOnOnOnOnOnOnOnOnO 




« -^-vO 00 CM Tf-NO 00 CM ^vOOO CM *vO 00 CM «*vO 00 CM ^-vO 00 Q 

hi hi m hi hi cm cm cm cm cm rofomcom^^'tNf^uNioinin mv© 


j 



274 



FUNCTIONS OF A ONE-DEGREE CURVE. 



O Ci ■tiflSO\H N -<1-MD NOh C) -^-"O t^ On h IN M-vO t-» On m CN "^-vO t^ On H 

ui 10 "i in to io\o vovo^ovovo t^r^t>.t-.t^ t^.oo oooooooooo o\om?»om>0 



to on cm to on « to on n toco cm tooo cm toco n inoo cm moo cm inoo cm moo 

OOHMHMNNrofOrO'^-Tt--<i-ioin tovo >ovo nn t*.oo oooo onOO^ 

>0>0>ONO>^ONO>ONONON^O^C^<>^^0>ONONONON^^O>0>C7>cyiONO> 



CM n-lONO>0 cm •*»■ 

OOOOOHMM 



lONChO CM iJ-inKOO CM 'tlDNOO CM * lO MJ 

m t-t m cm cm (N cm cm cm ronromrofO't't'tTf'finn 



fO'+'+'+'*'*'+i)-THoioioin»omio iomd vomdvDvo^dvOvo t^t^t^t>.t^r 



6 

en 

o 


ON O H 


h co tovoco cm 

CO "*■ IOVO t^ ON O 


rJ-vO CO H CO lOCO O CM 

h cm co m\o t^oo o h 


lOOO O COVO On CM 
CM CO IOVO t^OO o 


tJ- r^ covO 

M CM ■<*- IOVO 


c 


COTt-Tj-Ti-ThTj-Tt-Tj--^-lO 


to to to to to to iovO vOvOvOvO^OvOvO f>« ^ t^ t-^ C-^ t*» 


£ 






N T»- t>. ON H tJ-VO 00 H CO 

•<i- t^ covo covo 0> covo 


lOCO 
ON CM lO 


CM tO t^ On CM 
On CM tnOO CM 


rf-yO On 
IO00 H 


H COnO 00 
lOOO M -^ 


co to f>. o 

00 H *NH 


o 

53 

u 


ON CM MD OiNVO 0> CM to ON CM mOP) lO 0> CM lOIJN lO On CM lO On CM >0 0>N mO 

o>OOOMHMCMN(NmroroM-^-rt-inin tnvo \ovo nn t^»oo oo oo o> o on 

t>.00 OOOOOOOOOOOOOOOOOOOOOOOOOOOO'XJOOOOOOOOOOOOOOOOOOOOOOOOOOOO 



FUNCTIONS OF A ONE-DEGREE CURVE. 



275 



m ro mvo oo h ro mvo oo O m ro »n\o oo oa ro in t^.oo 0) ro m t^oo n 
lO IT) ID in mvO vOvO^OvO^O t^t^t^t^»C^ t^.00 00 OO 00 oo co o> o> o> o> o> o> O O 

loininiomiomiomioioiominioininiflinioioinininioioiom mv© vo 



Q 

o 


oo •<*• vo N c?> m h sroO t^» ■<*- t^ ■<*- t-sTt-o N'tHOO mmo t>. t*- w o 
ro m t-^oo w ro mvo oo O m ro mvo oo m ro ino oo o h mm t^oo N ro 


Q 


\OvOvOvO t^t^t>»t^t^ t-«.oo oooooooooo OOOOOOO O O O OO H H M 
NNNNWNNNNWNNNNWNNNNWNWrororororocorororo 



•* t>. rONO ro^O ^NvD O^M moo m u->00 h -* f^ ■<*• t». rOvO On ro^O 0> 



m ro Tfvo 00 Onh co -<*-\0 00 O m ro ^-O 00 0> m ro tJ-vO 00 0\ h ro »nvO 00 m 
OOOOOO^wi-iMHMfNONNNNOjrorororororoTt-'^^-^Tj-iniO 



Q 

o 


o 


'ta'toN Tj-co 


rooo 


rooo 


rooo 


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ro loo oo 


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rorororororo-^--«*-Tt-Tj-Tj--^ 


.4. 


in in in in u*> in\o vo vo 


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in mvo vO vO 


t^ t»» t^.oo oo oo o o o 




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ro ro ro ro ro ■*■ 


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ttiomiftui mvo 



276 



FUNCTIONS OF A ONE-DEGREE CURVE. 



55 



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o 

u 



fc 

S 



i m n w N w cm rorocororO'<*-'^-'^-'^-'*inioioiri m^c 



w "^-vo t^ On m ro -*J-vo oo On m ro -<i-*o CO Oih ro -«l-vo oo m ro m\o oo h ro 
inmioin m>o vco^ovovo t^^.t^c>«t^ t^.00 oooooooo on o> On on o> o> o O 



W N N N W CO 



COvO On N IO00 N lOOO H i-NO rOvO On r0>0 On<M moo H ^- S +N0 rONO 

SO tONO MNO MNO ro t>« rovO rovO rovO rovo rovO rovO 
OOOOHHMONNmromTj-Thrhiou-) m\o vo vO s s sco oooo aao 
N corocoMmcofOrtmrorofommfOfOfOfnforomrorofOromfOforofO 






£ 



o 

Q 



2 

£ 



I OnOO MO-tfON H O\00 S 



N tom S00 O 0) ^ m t>. On N ^mtsaO N -Tf in t>« ON H N -*-\0 S On h N 

OOOOOMHMWMMNNNN(NNrnron-)fOrOfO'<J-^'<*-rj-Tf'<t-ir)in 
vOvOvOvOvOvOvOvOnOvOnOnOnOvOvOnOnOnOnOvOnOnOvOvOnOnOvOvOvOvOnO 



On N 


moo 


M 


to 00 


H 


•*N0 


MNO 


rovo 


0> N 


m oo 


N ir 


CO 


H 


Tt" S 


*NO W 


ON 
M W 




CM 


to. 

o 


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CM N 


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(N M 


ro 
IN 


ro 

N 


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ro M- 


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r-. O 

Tf lO 

cm cs 


^-t^O -*-r^O ■<*- s o ■*- s ■*■ t>» 

ID in^O ^OvO NN S0O 00 00 On On 0> 
CMCSCMCMCMCMCMCMCMCMNCSCMC1 



FUNCTIONS OF A ONE-DEGREE CURVE. 



277 



t^t^c^t^c^c^t^c^r^t>.c^f>r^t^t>st^t^t^c^fxt^r^t^^r-%t^ c^oo oo oo oo 



On CM WOO w "*■ t*» O rOvO On CM WOO h -* t^ o rovO On CM WOO h ^- f^ O rovO 0> 
W ON CM W On CM W On CM WOO CM WOO CM WOO CM WOO H WOO w WOO H moo M t*- 

O>O>0 O O •"• <-* *-> cm cm cm ro ro ro rj- + ^- in in wno ^OO NN t>«oo oo oo On On 

■^■■<$-Lnu->WWWWWWWWWWWWWWWWWWWWWWWWWWW 



fc 

s 


O « -^-vO 00 CM "*vO OO N tJ-vO CO CM rJ-vO 00 O CM -^"O 00 CM "J-vO 00 O 
H M M M H CM CM CM CM CM COrOtOCOfO't't't^tinWlOm vovO 


Q 


roroeiCMHHHOOO>o>OON Onoo t^.vo uiiO't-fON h h o Onoo ts. t^vo w 

W CM 0\\0 MO t^^-M t^-^-CMOO WCM OnvO fO O N ■+ H 00 WCMOO WCM OnvO CO 


< 


ro wvo oo cm ro w t>»oo O cm ro w f^oo cm -*■ w t«^ on cm i-wf^o>0 cm tj- 

OOOOHHHHHHWNNMMNforotonnm'ttTj-Tf'ttininui 


u 
w 

C/3 


ro ro Tf WO t*»00 On M CM ro tJ-vO t^oo H N * WVO 00 CM tJ-^O 00 O CM Th 
CM -^-nO OO CM Tj-f^ONH rOWt^ONH -<*-nO 00 O CM tJ-\£) Om fOWC^O CM ■<*■ 




rororororO'*'<J-'^-'<l-'<*-wwirvto wvo vO vo vo t^ r^ t^ t>* t».oo oo oo oo On On On 



cm cm ro ro r 



NO On CM WOO M WOO H ■<*- f^ CONO ON CM vO ON CM WOO H Tf t>« O rO t^ O r0<0 On 

vO On rOvO On rOvO On rovO On rovO On CM vO On CM vO On CM vO On CM vO OnCMvO OnCM W 
OnonO O O "- 1 h m cm cm cm rororOT*--<i-Tj-ww wvO vO md r^ r^ t^oo oo oo on On 
rorO'l--<t-^-Tj-^-^-T*-^-Tj-Ti--^--<j-^.^-^ T )-^-^--^-T*-^-Tj-rt--<j-^-Tj-^^--^ 



278 



FUNCTIONS OF A ONE-DEGREE CURVE, 



y£> oo on m ro -^-^O co G\ h ro mvo oo m ro u-> t-^oo o cm ro 10 t-s.00 cm ■**■ io t^ 
io m iovo vo^o^ovo^o t^ c^ t-» t^ O.00 cocooococo u(MJiomjoo o O 
oooocooococooooooocooooooocooocococooooooocococooooo o on on on on 



h't't-t^mm uivo vo vo vo r>. t^ t^ t*»co oooooo ovOiOO o 



Q 


on cm moo h -<*■ t^ on cm moo h ■<*■ t^ o rovo 0> <N moo h rovo on CM moo H t(-S 


O 
X 

u 


ro t^ O fONO covO rovO O rovO O m\Q 0> rOMD On rovO On CM NO On CM vO ON 
OnOO O h m m cm cm cm romrO'+'+'+M-min »/->no vo^O nn t^»oo oo oo o« 





1 CM 0>vO ro 
} f>.00 CM muNNOiO CM 'tlONOsH CM -^j-vO SO\H N -*J-vO 

j u i-i « m h m m cm cm cm cm cm cm rororOfOfOro^->tt-t-t^inirii^in 

3C»00COCOCO00CO0000COCX)00COCO00CO000000COCOCO0000C0000000C0 



irjvO 00 CM ' 



>n cm inoo h •>*■ t^ cono 0> cm toco h *no rovo On cm moo h ^ t^. o roo O 

hOO h Tt-00 h -tf-00 h tJ- f^ h rj- t-^ w ^-t^M rl-t^O -«*- f^ O ^r^O tJ- t^ O ro 
IN ON O O O H H M CM CM cm mroro-^^-Tj-iou-) mvo vo vo t>. t^ f^CO 00 CO ON On 
■> mvo vOnOnOnovOvOnOnOvOnOnOnonOnovOvOvOnOnOnOvOnOvOvOnOnOnOnO 



CM tJ-vO CO O I 



FUNCTIONS OF A ONE-DEGREE CURVE. 



279 









OOO N ^-inNCJiO N -3-vO t^ O H N Tt-vO 00 O m CO ">*-vO 00 h CO lOvO 00 O 
lOvO vO vO vO vO vO C>«t^t^l>»t^ t^OO OOOOOOOOOO OOOOOQ O Q O Q w 
O^^O^ONCJi^^^^CTi^O^ONONO^ONOsONCJi^ONCTiO^O o o o o o o 



moo h covo o> n iooo covo (>« mso covo <y> n ■<*• t^ o coo on m ■>*- 1>. 

H -3-00 M -3-t^w -3-t^H -d-t-»0 -<*■ N« 4->-0 ro NO fOSO COvO COVO 
ON ON On O O O m >-< m o* OJ N cocOCO-3--3-'!l-iriir) iovO vO vO C-» t-» NOO OO OO On 
OOO0O0 ONONO^OnOnOnO>0>ONOnOnO>ONOnOnONO>ONOnOiO*OnO>0>OiONOnON 



2 



z 

s 






u-> M OnvO CO N* ^- W ONvO CO C 

t^ ON N -3-vO t>. 0> H N -<fvd 00 ON M CO -3-vO 00 m CO lOvO 00 O N fOW N00 
o h h m m m m w n n w w cm cococococo->i-"<*--<*--<*--5i-Tj-u->miou->ir)u") 
ONONONOnONONONONONONONONOnO*OnONOnONONONO>ONO>ONOnONOnOnONONON 



& 



p 

o 



t-» COVO C\N inSO COVO ON W 1000 h *S0 COVO Oih -tSO COvO ON ifl 

WvO 0\ N m^N lO ON W IO00 N lOCO (N IO00 W "">00 H IO00 m IO00 H tJ-00 I 
O0»0\0 O O h m m N <N N cococo-3--3-->*-ir)U-) lOvO vO vO N N. N00 00 00 On 
t-» t>» N00 oooooooooooooooooooooooooocooooooooooooooooooooooooooo 



55 



280 



FUNCTIONS OF A ONE-DEGREE CURVE. 





H 


55 


N Tt-vO 00 M -*vO 00 <N -^-vO 00 N "«*-VO CO N TfvO 00 N ^vO 00 Q 

m h h h h in m N in cm roM«com'<J-^"+'*'+inininio rnvo 


< 


O S*« OnvO "tHOO in CO S-tN OnnO CO h 00 in CO S^« OnnO fOHM 


N miO 1^.00 O N ■<*- in t-^ On H d t)-vO NOh PI «i-vO CO O H CO invO M O « W 
vOvO"OvOvO t^t^t^t^t^ O.00 COOOOOOOCO On ON On ON On O O M H M 
OOOOOOOOOOOOOOOOOOOOOOhhhhmhmhm 


u 

W 
CO 

X 


• OO h CO -<i-vO OO On 
lOOM uioo m ^-t>.H -tNH ■<*• t^ fOSO COVO covO On COVO On conO C/n IS 


r-v t^co coco oo>o\0 O O w m h <n n 04 cotoro*<*"+*inifl mvo vovj s 
OnOvOiOnOiOnOnOnOOOOOOOOOOOOOOOOOOOOOOO 


Q 

o 

Q 


invO vo r>. t^co OnOnOOQOOO 


in>o <o vo t>. t>. t^oo oooo on on on o> o O O m h h cn n n mrom^'t^tm 
OnOnOnOnOnOnOnOnOnOnOnOnOnOnOOOOOOOOOOOOOOOOO 


Q 
« 
O 
X 

u 


»D00 O fOvO ON H -^ t^ IN IDOO o covo 0>H Tf t^ N »O00 H COVO o> H ^- t-. 

oo h moo h -5J-00 h <<j-co h «i- t>» h ^-t^o ■* t^ O M-c^.0 roNO covO covo 
oo on On on O m h h n n n mroro-ft'+ioin invo vovo nn c-«oo oo oo 

OOOOHHHHHHHHMHHMMHHI-IHMMHHHMHMtHW 

««NN«NO*NN««N«NN«NNNWN«NNO»e)«e>»«e<N 


s 


vN -<*-vO 00 N ■+VO00 ON ^4-vO OO N tj-vO CO N «*-vO 00 N -*vO CO O 


* 

© 


55 


N -^-vo CO N Tl-vO OO O N tJ-vO CO <N -«i-vO WON tJ-vO 00 <N t}-nO OO 

m m m m m o N cn w w rororofOfO'tirt't-tiominio mvo 


Q 


M-MOO m « S'tMOOvO COO t>. «*• H ONVO CO 00 in <N On N "+ h 00 vO CO 


O « fomNONO w ^lot^ONH n tj->o nonh ro -<*-vo oo o> h co m^o oo o n 
h m m m h h n o» cm n cm cm rootoforom^-^^^'t'finiflioin m\o no 

ooooooooooooooooooooooooooooooo 


u 

w 

CO 

4. 


On On On On On 0> On On On On m m N N m-^^-ir) \r>\0 00 ONO h ro + irjvO t^CO 
rovo On w u-noo h •<*• o. o ■* t^ rovo On w tnco h t*- t^ ">*• t*H rovo on n io 


000000 OnOnOnO O O m h h n w <n n ro m ro •<)■ -t 't in in iovo ^OOvO NN 
O000000000CO OnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnONOnOnOnOnOn 


Q 
K 
O 

Q 


in -^- ro « w o onco t^vo vo ininioinm^*'*^ , '* - ^ , ^"*^'*^-^-^-^i/N. 
cono on n m t^ o covo on w moo m •*■ t^ o covo on (N moo m -*- ^ o co>o ON 


t^ t>. t^ f^oo oooo OnOnonOnO O h m h w n w n cococoTfrt-^i/jininin 
COOOOOCOCOCOOOOOCOOOOOOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnO.OnONOnOn 

1 


Q 

O 

x 
U 




covo On covO on covo onn>o onnno Onoj ioonm inoo n moo w moo m inoo 
OnOnonOnO h m h n n w cococo-<i-Tj--<i-inin mvo ^oo nn r^oo oo oo 
OnOnOnOnOOOOOOOOOOOOOOOOOOOOOOOOOOO 

HHHH(4N«C4n(IOINNnNOIOIC4«NflC<nNNCICICin«C« 


2 


N ^VO 00 N ^-NO 00 N -^-NO 00 O W "*vO CO N •*vO CO CJ -^NO 00 Q 

m m m m h w o cn (N <N rorocofOcONj-ij-^-'^^-wivunin mvo 





FUNCTIONS OF A ONE-DEGREE CURVE. 



281 



CO 


S5 
3 


N -*vO 00 N ^vO 00 


N "«*-vO 00 N -<J-vO 00 N -^-vO 00 o n -^-vO 00 




00 iO« ts. ■* « o\ t^ -^ 


h a>vO -^-hoovo roHOO iflcoooo in « o mon o 


mso\H n tJ-vo m>h m tj-\o oo <-> ro in c-^oo o n -^-mt^oM w -*-^o oo 

vO vO "O t^»t^t^t^t^ t*»00 OOOOOOOOO^aaOM^OOOOOOOHHHHH 






u 

w 

X 


-^ r^ m -too h inoo w m 


onn^o a» m\o o cinh -^-oo m inoo n \o o\ co\o o 


t-N C^OO 00 00 <J\ <3\ <J\ 
MHHMI-IHI-IHNO) 


O h m m n n rorofO^'+'+mm invo vovo s t^oo 

««NNNNN<NNNNNCS<NNN01tNNNCi 




Q 
a: 

o 

Q 


mt^o -+N0 -«*-tN.H 


■«*• t^ m -^oo m -<t-oo m moo m moo N inoo n.i«o>ci 


ihhi-imhhmi-iwmhmwmmN«WNN(NW(NN(NNWMN(NN 




« 

o 

X 
U 


00 M tO^O 00 M -»J-vO CT> M 


M-NOiN ■**- t>» o N in t> o fO moo covo oo h covo 


ooco C\ on o^ o O m m m cm n n romro^'t^uiioin in\£> ovO sn r-^oo 
n N cs cs w mfOrofomrommforomromfOfOfomrofOMmmmfomm 


55 


« +o 00 « ^VO 00 

M M W M M 


<N ^vO 00 N -^"VO 00 N "*vO 00 N ^-vO 00 

cs n w 04 n rorofOcorO'^-'^'^-'^-'^-mmmm m\o 




2 

2 




N ^-VO WON "4-vO 00 N "*"0 00 N "*"^0 00 O 

cs w in n n forococorO'tt't^^inifliflin mo 


< 


oo m m o t^ •«*• in owo ■«*• 


hoovO roo t^mw ot^-^-M os^o *hoo\o roooo 




H N tJ-vO 00 On M CO -^-vo 00 M fO in t«»00 N ■*• m 






u 

u 
w 

. X 




OOOOOOOOOm 


OO h h h w n n mmwt^min mvo >o^o ss 




Q 

o 

Q 


nvo o\« moo h ■<*■ ■>. o 


■^■t^O '^■t^O *N0 *NO MNO MNO fONO 


u-> m invo vovo ss t>»oo 
OOOOOOOOOO 


oooo <J\ <J\ O O w h h n N n rororo^-^'^-m 
OOOOOmhhmhmmmhmmmhhmm 




Q 
« 
O 

u 


t>» N inoo cOvO On h 


•«*• (** <j\ <n m tx o roo oo h m-vo o> oj >*• ». ro moo 


vO rovO 0\ cOvO Q> N vO 

ooa.aNa>aNOOOMH 

NNNN0J0JW0JN01 


o>n mo>N \n o\ a moo n moo m moo « inoo w Tt- 
w « n cm mrorOTj-Ti-Tj-inm mvo ^o*o sn f^oo oo 

N<NNNNN«NNNtNNNNN(NN«N01fN 
MN0JNWCJWeNNNWC4WW(NNN«(NC»<N 


si 


« ^-VO 00 N "J-vO 00 
H H H H M 


N "*vO 00 N -^-vO 00 N ^"O 00 « ^vO 00 Q 

w n « « N ficommnt** + 'tifl>nio« mvo 



282 



FUNCTIONS OF A ONE-DEGREE CURVE. 



< 


ro oo tnroooo iomhoovO mhoovO ro m oo vO ro h oo vo -<f- h on<0 m- w On 

cm ro»ONOO cm "<*-vo so\h ro m-nO oo O m ro m t>»oo 6 CM -tmso\H cm 
NNtMsts t^OO OOOOCOCOOOOnOnOnONOnOOOOOOmmmmwi-iCMCM 
CMCMCMCMCMCMCMCMCMCMCMC<CMCMCMCMCMrorocorororororororororororo 


u 

M 


H in ON COvO "<*-00 CM iOO\rONH tO ON COvO "^-00 NVO O "^-00 N lOOrON 


in 

w 


o> on o> o m m m cm cm n rofO"<*-Tt-ifir> \n\o ^oo s i>.oo oo oo o o> on o o 



00 CM lO ON CM VO COt^O "*00 H IO00 N vo CO t*» H inoo CM in On ro t^ ^-00 
mvo vO vo tv t>»oo oooo O\0\00 O O m m cm n n ro ro ro "*- rj- -<}- to mvo md \0 

rj-vO On M -<i->vO On H -^-vO ON H tJ-vO On m -t}-\0 O w ^*"VO ON H -«l-vO On H -<**vO On 

rovo rovo On CO^O CfHO Q\ M U"> ON CM lOOO CM u~>00 M IO00 H tJ-OO ih -<J- t^ 
oooooo o> o> On on h h h cm cm cm mmM^'t'tioin mvo OvO nnn 

CMCMCMCMCNCMCM<NCMCMCM<NCMCMCMCMCMCMCMCMCMNCMCMCMCMCMCMCMCMCM 







00 


t^ ■>*• CM ON t-* "*■ 0) 


OnvO •<*• w On^O 


rf 


hoo^o ro m oo vO rOMOo m ro oo in m 


on m ro x*-vo oo 

H CM CM CM CM CM rO 
CM CM CM CM CM CM CM 


m ro in tv.oo O 
ro ro ro ro ro tt 

CM CM CM CM CM CM 


CM 


tH^noh cm tJ-O co On m ro in^o oo 
Tt-Tj--^-Tfir)ioiouoio m>o ^)\0\0^0 I s * 

CMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCM 




u 

w 



00 


-^- t^ M \O00 CM \Q 


On ro t^ -"i-00 




in On CM <0 rJ-t^M in ON CM VO -<*-t^M 


C/3 

x' 

w 


00 00 ON ON ON O 

cm cm cm cm N ro ro 


M w CM CM CM 

ro ro ro ro ro ro 


e' 
rr 


ro ro ■*■ n- in in mvO ^O^O N t-^oo oo oo o> 
rorororororororororororororororo 





CM in on CM vO On en\Q rONO -<j- f* w "*00 m ioo(^ ^ vO On rovO O 



•^-«0 ON M Tf 



vO On m ■«*- > OnCM •*}- f^ On CM in N CM moo CO lOOO i 

v. IO ON CM lOOO CM lOOO I 
000000 OnOnOnO O i 

rOfOrOfinrO^'^'t'T'T'T'T'T'TTT'TT'T'T'T'T'TT'T'TTT- 

CMCMCMCMCMCMCMCMCMCNCMCMCMCMCMCMCMCMCNCMCMCMCMCMCMCMCMCMCMI 



FUNCTIONS OF A ONE-DEGREE CURVE. 



283 









o * 


55 

2 


N ^"O 00 N -^-vO 00 O N "*-vO 00 N M-vO 00 N -<i-vO 00 N ^vO 00 

m m m m m n cm cm cm cm rofomtorot-t-t't'^ioinioir) mvo 




VO "«*- CM OMAN 000 in M 00 vO MM OvO ■* CM O*MnfO000^O Mm OwO 


lflNOvO N -<4-vO 00 On m ro invO 00 O CM muiNO>0 CM *<a-vO 00 O m ro mvo 00 
t^ t-» tN.00 OOOOOOOOOOOnOnOvOvOvOOOOOOmmmhimmCMCMCMCMCM 




u 
U 

m 

X 

w 


00 CM t^M irtO^rONH in ^00 CM vO m in On MOO CM vO m O M t^ M vO ■*■ 


CM MM'tTf^in lT)vO vO NN t^CO 00 On ON ON O M M CM CM CM MM^TJ-iniT) 
vOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvO t^t^t^t^t^f^f^f^t^t^f^t^t^ 




Q 

o 

Q 

i 




0000 O*O\»0 O m m m CM CM MMM^-^iniO. mvO vO NS t^00 00 (J\ On O 
l/l in in in invO vOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvO (■>» 


« 

o 
s 
u 


M lOOO CM iflSO CM ^t^OM <^-vO 00 M MvO 00 MlONO CM WSOnN ^* 


mOO M m00 M -^00 M -*N0 "*-t^0 M f<» MVO O MvO 0> MvO On CM in On CM 
t^ t^OO 0000 On On On O O M M M CM CM CM MMM^Tj-Tf^-inm U">vO VO^O N 
vOvOvOvOvOvOvOvo t^t^t^r^t-»t^t^t^f^t>.t'^f^t^t>.f^t^f^t^f^t-»t>.r^t>. 
CMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMeMCMCMCMCMCMCMCMeMCMCMCM 


55* 


CM ^vO WON -<*-vO 00 CM "*vO MOM <<J-vO 00 N ^vO 00 CM "*-vO 00 

M h m m m cm cm cm cm cm nconnntttttminmin mvo 






CM "*vO CO N ^-vO 00 CM -<*-vO 00 CM -*vO 00 CM "^VO CO « "J-vO 00 

m m m m m <n cm cm cm « rotflMmcO't'ttttiominui mvo 


Q 
< 


OnvO ■* M ONVO ■* CM OSN-+N t-x -<1- CM MOM 000 in M M 00 VO ■* m ONVO 


cm ^t-vo oo o> m m invo oo cm rowsovH cm -^-vo oo o> m ro mvo oo cm m in 
cm cm cm cm cm roroMMMTj-rj-rt-Th^--<*-ininininin mvo vo vo vo vo t*» t^ f>» t-* 

MMMMMMMMMMMMMMMMMMMCOMMMMMMMMMMM 




u 
w 

c/2 

X* 

W 


O M M m CM CM rOMM^frJ-inin invo vO NS t-»0O OOOnOnOOOmmNNN 

ininininininininininininininininininmininin invo vo vo vo vo vo vo vo 




Q 
PS 

O 

Q 

■a 


oo h moifONH tj-oo NvO -<i-r^M iaonmnC ^oo nvO Mt^M ino>M 


>o t>» t-» t^.oo ooo>ONONOOMMMNNNMM^-*^-in invo vo vo t^ r^ t^oo 
Th-^Tj--<i--^--^--^--^--^-inininininininininininininininininininininin 




Q 

OS 

o 

K 

u 


On m -<i-\0 0> m -<J-vO 0> m -*vO 0> H ThvO OO m MvO O0 m MvO 00 m MvO 00 m M 

^»m 4-t^d 4-t^0 Mt^o mvo mvo o> mvo o>Nvo o>n inovcvi moo w m 
t^oo oooo ononOnO O m m m <n n w n roMM^-^^inm mvo o^ nn 

in in in in in in invo vOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvO 
WNNWNNNNNWCV1NWCV1NNNWNWNCMNWCV1««WNNN 


55 


N "^"VO 00 N "^"VO 00 N -^-vO 00 N ^vO 00 N -"4-vO 00 W -^-vO 00 

m m m m m n w n w cm mconcotottti'tioi'ii'iio «ovo 



284 



FUNCTIONS OF A ONE-DEGREE CURVE. 



i m m m cm cm cm cm cm rorOfomrO'*'*'*'t'*ioioioin mvc 



O t>s »*■ cm OoovO com ONf^mw OoovO ^-h osinMOoo^O ■<*■« N»nm 

H C0 10M3\0 cm "4-vo SOh fO lOvO 00 O « wms6\H N -4-vO 00 H CO vn 

00000000000>0»OOMMJOOOOOHMHHHH««NPl«fOroro« 

\r> ■*• O moo N Nh mo iflO> -+00 roNN shvo m(> "4-00 rooo CM t«> h 

CO O\O*O»0 6 h h CM CM CO CO CO "*- ■<*- in iovO vO t^ t^OO OOOOOnOnOOmihCM 
00COCOCO 0(M>Oi(M>OM>(MJ(MM>(M?OMJ(>(>0>(M^O O 

MMMHMHHHHHHHHM*I-II-IH»HHHHMMHHCMCS)CMCMCM 

in On CO ^ CM vO "4-00 COSH mO -^00 CM tN. H \r> Q\ rj- On ^00 CM VO "4-00 CM 
w cm co co -<f "4- io in mvo vO t^ t>»oo oooo o> On h h cm cm co co •<*• -<4- "4- in 

OOOOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOO 0\a»0>ONOONONOOOM>0\ON 

■>4-vO 00 M CO in (^ CM "4-vO 00 H fOlONO CM "<l-vO o\m m lOOO CM "4-vO On m 

On CM in On CM li")00 CM mOO H -^-00 M ■<$- t*s M Tt-r^c5 co t*«. comd covo On cm nO 
VO NN t^OO 0000 On On On O O O h m m CM CM CM mfOrn^'+'^vninir) lovO vO 
OOOOOOOOOOOOOOOOOOOO OnOnOnOnOnOnOnOnOnOnOnOnOnOnOnONOnOnOnOnOn 

CMCMCMCMCMCMCMCMCSCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCM 



CM "*VO 00 CM "4-vO 00 CM "4-VO 00 CM tJ-VO CO CM "4-VO 00 O CM tJ-vO 00 O 

h m m m m cm cm cm cm cm rofOfomroi i '*'*'*'t»oinioifl mvo 



O CM "*-vO 00 O CM <«*-vO 00 O CM -^-vO 00 O CM "4-vO 00 O CM ">4-vO 00 O CM <<*-vO 00 Q 



< 


VO "*" CM NinrOHCOvO "4- CM On t-x in CM OOOVO fOH OsMOM OOOVO rOH o 

co c « 4-msoNH « Tt-vo co o\h mm r^.06 o cm ^msoNH co tj-vo 00 o m 
cm roMromcoro't'<l-'t'+'t-+>o»nioio u->vo vo vo vo vo vo r>»t^t^t^ c>.oo oo 





•^•00 roNHvO O ThONrOt^CMvO O in o> cOOO CM vo m in o "4- On cooo cm vo h m 



Q 

OS 

o 

Q 


cm vo -*oo cm vo "4-oo roNM martSH inov tj-oo cm vo ttoo cm vo o m 

O o h h h o « rororo<t + mm mvo >o sn t^oo ooo\OnOOOmmcmcm 
t>.t>.t^t>.t>t^t>.t^.t^i^t^t^t^t^t^r^t^t>.t^t^t^t^t^ t-^oo oo oo co oo oo oo 




Q 
K 
O 
X 

u 


■^■vO On h covo 00 fimsO CM +SOm -<4-vO 00 h n inoo CM to t^ o> CM Tf 

cm inco cm moo h moo h -<t-oo m -tso ■<*- t*» mno covo covo o cm vo on 
r^ t^ tCoo OOOO OnOnOnO O m h w cm cm cm rororo'ft'tiomm iovo vO vo 
r^ t-» f^ r>» ti. ti. t>. t^ t^oo oooooooooooooooooooocooocooooooooooooooooo 

CMCMCMCMCMCMWCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCM 



FUNCTIONS OF A ONE-DEGREE CURVE. 



55 


N ^VO 00 <N »*-0 00 <N «*-O00 N ^O 00 N -^-vO 00 CM -^-vO 00 Q 
H h m h m cm cm <N cm cm cocococorO'«i-'«-'<j--«-Tt-ininioin ino 


Q 


00 O ■* CM 00 O <*- CM OOOvO rt-N OOOvO -^-N 00 O \T) CO H OnNIOMh 


On O CM "^"O 00 On H CO in tN.00 CM -*vO t-^ On M CO mO 00 CM *mSO>H CO 
OOONONONOiONONOOOOOM>-iMHi-ii-iCMNCMNWcococococorOTj-Tt- 
iO in in ui m in inO OOOOOOOOOOOOOOOOOOOOOOO 


u 
u 

X 


cooo cm c^ cm N«vO ho h\0 mvo m m mo »n0 m ^- On "^- On -<l- o« 


\00 s r^oo ooONONOOCMwHCMroroTi-^-m ino O t^ t^oo oo oo on on 

MMWMMi-iWMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMNCMCMCNCMrOCO 
CMCMCMCMCMWCMWCMCMCMWCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCM 



•<*-oo co t^ cm o h mo -<J- On cooo cn t»» m o m On "*-oo mnnvO h mo m 
OOOOtMJOOHHNNNMWttin ino OO N t-^oo OOOnOnOOhmN 

OOOOMMMMI-IHMMMMMMI-ll-lMMMMMlHMtHMNNNCM 

NCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCM 



vo oo N Thvo oo ro in t^ On w mmso^M roms»M -«-o oo cm -^O oo 

« mONN moo m moo m ^-nh ■<*• t>» o mt^O mo o roo o> m o o cm moo 
vooo ss c^oo oooo o>o\on0 O O m m t-> n cm n cm mfOfnTj-^-Tj-inmin 
OOOOOOOOOOOOhhmmmmhmmMmmmhmmwmm 
rorofnfnrortrocncnforocnmrnfnforofnfOforofOmfnfororofnfOfnro 



ro m OMnnOooo -^-N Oooo ■**- cm smrOH o^MnroH i^Mnw 

m tN.oo o cm -^-o t^ on h co ino oo cm "tmsovn cm -<i-o oo o m ro m f^ on 
fOfom't't't't't + ininioifl mo ooooo r^t^t^t^ t^oo oo oo oo co oo 
inmmmmmmmmmmmmmmmmmmm\nmmmmmmmmmm 

mo m mo mON-^-ON rooo ro t^ cm t«.MO h mO mo ^- On -«*- on -<*-oo rooo ro 

cm cm rpro^-^Tj-in mo* O t^. t^oo ooo^O^OOmmcmcmcmcoco^--'"-^ mo 
OOOOOOOOOOOOOOOOOhmmhmmmmmhihmmm 
CMCMCMNCMCMCMCMCMCMNCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCM 



cm o m in o •^■oo cm o h in on -<*-oo ro t-x m o m Q\ rooo CM t^H mo "i- On -* 

m ino o t^ t^. t^oo oo OnOnonO O w m cn cm ro ro ro tj- t*- in mo O t^ t^ t-^oo 
OnOnOnOnonOnOnOnOnOnOnOnOOOOOOOOOOOOOOOOOOO 
wmmmwi-immmmi-ii-iCMCMNCMC'IcmCMCMCMCMCMCMCMCMCMCMCMCMCM 

** ro ir> <*>» w -«i*-o oo O rtmso\M coo oo cm -<*-0 On m mmsoN ^i-o 

o on cn in on cn moo m moo m ^t^w T^-r^o 4-t^d roo coo On cn o on cm 
o o t^ t>> r^oo oooo OnOnonO O m m m cm cn cm ro m m * t * * m m mo 
ONOnOnOnOnOnOnOnOnOnOnOOOOOOOOOOOOOOOOOOOO 
cn cm cm cm cm cn cn cm cm cm cn rororocorococorocococococorocorococococo 



286 



FUNCTIONS OF A ONE-DEGREE CURVE. 



mh On f^ 10 -*• cm O oo vo 10 co h on t>.vo -»i- cm OoovO mrOH o t^vo i*- cm o oo 

N0\0 N -4-vO CO H CO lO t^ ON CM "tf-vO CO m Ml«SO\6 CM 4-VO CO 6 m 
a\CT»OOOOOMHHMHHNNNNNmromrororo-<J-'<*-'<*-'<*-Tj-u-)ir> 
vo vo t>»t^t^t^t^t>»t^t^t-»t^t^r^t^t^t^t>»t^t^.t^t^t>.r^t^t>»t^t^r>.t^t^ 

HVO H f^ CM t^ CM t*» CM 00 COCO CO ON ■«<- ON -*1- ifl O m N VO H VO CM t>. CM 00 COCO 

«3vC5 s t^co oo (MJO O h h « n nro + m mvo vo t^ t-^oo co on On h h 
r^rj-^-Tt-Tj-'^Tj-^ininininminmininininminminininin in\o vo vO vo 
CMCMCMCMCMCMCMWCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCM 



"<*• ON "*■ ON •»*■ ON cooo COCO CM t>« CM t>. « VO H v© H vO 1O0 io "*h ON ■<}■ On ■*• 
VO VO vO N« t-^00 OOONONOOHMCMCMCOCO^Tt-in mvo vo t^ t^OO CO CO ON on 
CMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCM-CMCMCMCMCMCMCMCMCMCMCMCMCMCMCM 



On m ro in t^co cm Tt-<o co cm -*J-vo t^OH mmNOH co iovo co « -<*-vo 

4-co m 4- t*> O ■*■ t>. O covo covo on cm in on cm ioco m tnco h -*■ t>» m tj- !■>. o 
in invo vo^o sn t^co coco on on on on O m h h cm cm cm co co co -4- tj- -*j- m 
cm cm cm <n cm cm cm cm cm cm cm cm cm cm cm rorotofonfofonnfoconrnronn 
ntocoroMfOcomMCiOrocotOfOfOrorOMcofomrorocofororocoMco 



h UMflro 



I ON f^ m CO H Q\MflrOH ON t^ m "<J- CM OCOVO *<*- CM OCO MOfO 



co "^vo co cm co in t^ o> m cm Thvo co o h roioNOH cm tJ-vo co o m en in j^ 
****ioinmiom mvo vo vo vo vo t^t^t^t^r^ t^co cocococo on on o> on on 

>OvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvO 



ON"$-ON"*'ON''*'ONTfON''t-ON'<t-ON'<*-ON«*-0 lOO ir>0 m lO lOHVO HlO H 
OHHNNcorO'^'^-in mvo vO t^ t^oo o> o> O m h cm cm co co -<j- t$- m invo 

CMCMCMCMCMCMCMWCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCM 



Q 

o 





m 


o> "*-oo 


COCO 


CM 


t>» 


H 


VO 




m 


^ 


ON tJ-CO 


CO f-. 


CM 


t"^ CM VO 




in 


lO 


in 




p5 






^f. 




t>» t^.00 CO CO 


ON On o 





M 


H 


rM 


CM 


co co •<*- t)- in 


mvo 














CM 






(N 






CM 


CM (N 


CM 


CM 


CM 


CM 


corococorococorororococoro 


Q 


CM 


CM 


cm 


cm 


CM 


CM 


CM 


CM 


N 


CM 


CM 


CM 


CM CM 


<N 


CM 


CM 


CM 


CM 


CM 


CM 


CM 


CM 


CM 


CM 


CM CM 


CM CM 


CM CM 


£ 

























































00 CM "<*-vO 00 CM -^VO 00 CM rJ-vO OO CM 



NNO\H COlOt^ONM COlOt>.ON 



cocococococococococococococorocococococococococococococ 



FUNCTIONS OF A ONE-DEGREE CURVE. 



287 





h 

CO 


z 

S 


N -^-vO CO N -^-VOCO N -4-vO CO N rJ-vO CO N -«*-vO CO N -*J-vO CO 

m m m m m n n w w n rtMromfO't^'t-t'tinmioio invo 


< 


MflfON 000 tslOMIN OCO KmrON OCO t^lO-^N H 0\ tN.vO *MH OCO 


VOCO N "tWM>H m U")<0 CO W "*vO NO>H MlO fs.CO N ^J-VO CO On M 
O O i-i m w h hi w cs N N 0) N rorocomror<">Tj-TfT$-Tj-Ti-ir>u-)U->u~>u-> m\£> 
COCOCOCOOOCOCOCOCOCOCOCOCOCOCOCOGOCOCOCOCOCOCOCOCOCOCOCOCOCOCO 


u 

w 

X* 

W 
I 


H vO M t>. COCO ^OMflOvO N SM^'tO lOH N«O0 M- O in vO N SMO> 


coco ooo o •-• m n co ro ^ tj- in ir,\o t^ t^-co cocr>ooOt-iNNrocnTt--»f 
snn r*»co cococococococococococococococococo OOOOOOOOOl 


Q 
K 
O 

Q 
2 




in invO vO t*» t-^00 COOOOOwwNNroroif"*-^ io^o vo t-> t-^co CO o o 
\OvOvOvOvOvOvOvOvOvO t^t^t^t^t^r^t^t^t^t^t^t^t^t^t^t^t^t>»r-» t^CO 


d 

K 

O 
X 
V 


m ro mvo co N towiso^O N -^vo I s * O m co invo oo O N nmsoo N * 


vo O N inco n inco m tj- t^ m -*■ t^ o rovo rovo On n vo o w inco m inco w 
■»*• «4- m m mvo ^vO sn t^co cocoooooOOOMMMNWNfOcofo^ 

^■^■^■^■'+'t*^'t'*'t't*'t , i-^ , *-ioinuiinmminmini^inir)inin 
nrorOfOrorOfOrorororomfOforofonroroforororoMfOfommrorom 


2 




i 
1 


S3 


N ^VO CO N "«*-vO CO N ^-vO CO N <<J-vO CO N -*-vO CO N "*vO CO 

h m m m m n n w w w rocoroMco^^^-'f^ininmin invo 


a. 


CO MflfOH CO vO *fOH 0> t-^vO "*■ N (>Mfli(-N 0»MflM« 000 t>» 


h miflscAH e* -^-vo co o h fomsoH n ^j-vo co o n ro in t^ o m co M-vo 
u-> in in in invo vo vo vo vo t^t^t^t^t^ t-^co cocoooco t>o^OM>oio>0 O 
^^N^^^^^N^^tsN^tNt^^^^sN^N^^^sMs^s t-»co co co co 




X 


CO ro O "*• *n vo m t*» oj t^ mco ro o> ■* *n vo h t^wco ro o M- *n m 


h n n co ■* ■*• m mvo vo t^ t^co co ooo h m n n co ro ■* -<i- in mvo t». t^co 

vOvOvOvO^OvOvOvOvOvOvOvOvOvOvO^O t^t^t>.t^.t^t^l>.^^t».t>.r^r^t>»^ 


Q 
K 
O 

P 


■^-00 N tNiN t^N t^N t^N l^N t>»W t»»N t^N t^N t^N ts.N t^N t^N t>^N 


o « h n « tomt'tm mvo vo f^ t^co coo>OOOMHNNmro-^--<j-in 
iniomminmininininmininininmininin invo \0"OvovOvovOvovovOvo 


Q 
« 

O 

u 


VOCO H fOlONO^H N rJ-vO CO W COint^OH CO ^VO CO N ^lONOiH 


O rONO rovo (jcivo 0>« inco N mco m -^-t^H ■^•t^O rot^.0 rovo O N vO 
in in m>o vo^OO nn t^co ooooDOM^OOOHHHNNNforomm** 

(ococococococomrom<n(ocororococorororoeorocoroco<orococococo 




« "*vO 00 O M -*\o CO N "*v0 CO N -«*-vo 00 « "*vO CO N "*-vO » 
h w m m m n n n w w rof0fococov^*"'* ,| *"'*«n»n»n»n mvo 





288 



FUNCTIONS OF A ONE-DEGREE CURVE. 



COM OOO NW^N H OnOO VO lOfON OnOO VO lO -+ « H 000 Mfl-tfOH 

r^ o» m n -*-vo oo o <n ninso\M ro m*o ooo n -«-vo oo 6 h minso\H ro 
m m n o m w w mmmfifom't'+'t^-tioiriioiJi mvo >o vo <o vo m3 c-- 1^ 

OnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOn 



rooo •^-OvO N00 "*• m H t^ ro 0> in H NfOOMONOO -«J-0vO WOO •* vO n 
N N ro "*- -*■ in invo t*» t^oo oo On 0> m h w n ro «i- -4- mv6 vD t>. t^oo o>o>0 

HMMMHMMHHHHHMMWNWNNM<NWNCM(NNWNWNrO 

MfOcofOrofOromrofOfororocorotomcofomroroMmromcoronmro 



Q 
K 

o 

Q 


m vo m t^ N r^ rooo moi'+O\>fl0 in m vo m t^ n tv coco r ^ o\ ^ oma o *£) cm 


v£) VO t>« t^OO 00 On On O H H N fOfO-^'^-m idvO vO t^ t^OO 00 On On H H CM 
OnOnOnOnOnOnOnOnOOOOOOOOOOOOOOOOOOOi-ihi-ih 
N « cm cm CM cm cm cm corororocoforoforoncofOforofOrorommrofOcom 


Q 


ro mvO 00 On h cm ^i-vO SO*0 N ro mvO 00 h ro Tt-vo NO»h cm tj- in t^oo 


O 

u 


vo O CM moo CM moo h ■<*• i>. h -<t- t^. o rovo O rovo 0> CM mOO CM mOO M -<t- t^ w 
ro ro "*■ "<f **■ in in mvo ^OvO NN r^OO OOOOONONONONOOO'-iHMCMCMCMro 

vOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvO^O t^t-^t^t^t>»t^t-^t^r^r-> 
rororororororororororororororororororocorororororororororororo 



55 

s 


CM -*VOOO CM ^-VOM CM -xt-VO CO CM -*VO CO CM ^vO 00 CM -*vO 00 CJ 

h m m h m cm cm cm cm cm cnMrorofOTj-^^-'t^ifiioioin mvo 




OOvO lOfON O t>* m •<*- CM H OnOO nO m CO CM On t>»vO *fOH ooo t^ in ■<*• ro 


h ro m t>s o> h cm -^-vo oo O cm romr^osH ro invO co cm "*vo oo On h roms 
vOvOvO^O^O t^t^t^t^ t^OO OOOOOOOOOOOnOnOnOnOnOOOOOOhmhm 
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOCOOO OnOnOnOnOnOnOnOnOnOn 


g 


On rf- o no cm smoMnOvO moo cooiwhvo noo ■**• ioh scor^mH t^ro 


in 


■xj- lOvO MD t-» t^OO OOOiOOHt-tNNrOThTj-m mvO t^ t^OO <» On On O m h CM 
OnOnOnOnOnOiOnOnOnOOOOOOOOOOOOOOOOOOhmmh 
cm cm cm CM cm cm CM cni CM rorororororororororororororor<-irororororororo 



■*!• On ■«*■ On in toOvO HVO I 



SO SNM rooo roON-^-ON'i-O mo iohvo h 



OOwwwroro-^-^-io iovO vO t^ r^oo oo Ono>0 m m n roro-^-Tj-io io<o 

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOCOCOCOOO OnOnO^OnOnOnOnOnOnOnOnOn 



Tj-vo t^ONO N •tmsciO w ^ in t^co w ro in t^oo m ro mvo oo o h ro 

H •«*■ t>. *NO rovO On rovo On N moo N toco m rr t^ m tJ- f^ O rOvO rovO 
^■^■^•inio mvo vOvOvO ss t~xOO ooooONONO>OOOHHWNNCNrororo 
lOininininmioininioiointoiou-Nininin iovo vovovovovonomdmdvovdvo 
rororororororororororororororororororororororororororororororo 



h io m m m >o^ 



FUXCTIOXS OF A ONE-DEGREE CURVE. 



289 



2 


r> 


CN 


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■>*-0 00 





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tj-O 00 


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3 


a 




















04 


N 


w 


c 


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nnromro 




1 1 1 1 1 io m m m m 


Q 







0> t^O 


in ■* m 


M 





o oo 


MO^fON 




oo 


t^O 


in 


"*■ N 







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t-» m 




_ 


N 


^-O 00 


n 


N 


^4. 


kO 


f^ 


<J\ 


H 


ro 


u~ 


r^ 


r?N 


M 


m -^-O oo 


n 


N 


^- 


O CO 


O M 


m 










m 


m rr 


-•. 






-r 




■<*■ 




LO to lo lO inO vO \o \o vO 


r-« 




r^ 


r^ 


r^ 


t->.00 00 00 


< 


o 


n 





o o 


o 


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■ 


n 







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U 


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t) 


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D 





O o 


n 


n 


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M 


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N <N 


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M 


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M 


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N 


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N N 


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N 


cs 


N 


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N W 


o-i 


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t»* m O^vO N00 tJ- H t^ fO 0\\0 N (MOH00 "<*• m n m ^O m On in N 00 in ► 



00 OOvO ' 



ih n roro't't mo O r^oo oo ao O m n n mm*m ino t^ t-» 
■^-TT'^""ju)inininininininininininin inO vDvO\ovdvo^o , OvOvOO\ovo 



SNCO m On in O N NrOO*-tOvO 



t-^ mco *o mn nwco •<*- on in m o 
i w n m m tj- m in 



OOOOOOMNNrorO'*-^- ino O f» f^oo oo O O O w 
mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm 



ino oo o o n m -4-\o t^oo o 



- m\o oo o n m ^o t^oo o 



moo m -<4-oo h rfso mo mo o> n moo m moo m ■<*- t^ o nso mo o 
n w mmm-^--^-^j-mm mo o o o c^» t^ c^oo ooooonooooOmmhw 
cococooooooooooooooooooooocooooooooocooooooooooo On On O^ On On On On 
mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm 





On t-xO m m n h oioo t>.o <<h m n o o>oo o m + m h o o> t>.o m •<*• n m 


m -^-o oo O n -*o s(^h m in t-^ on w n "<*-o oo n "i-o nom romNU 

t^ C^ C^ f^CO C0000000000nOnOnOnO>00000»-ii-'mmhmNNNNN 
0>OnOm^OOO\0»OnOvO>OnOm>0\0 O O O O O O O o 


u 


noo "*-oo moo in m t^ m on in h t^^oo woo ^w t^m o>o noo tn n 


C/2 


OOHNNmm-<fin mo o t^oo oooNOOMMNmm^Ti- ino o t-^oo oo 
mmmmmmmmmmmmmmmmr*-Tj-^rj-'<j-^Ti--^-^T*--*-^Tj-<^-Ti- 
mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm 



Q 
K 

O 

Q 


n t^ moo Tt-ONinoo h nnoo moi-^-0 mno w t~N moo ■* o> m o o h t^ 


w n mm-*'* mo o t^ t^oo oo o>o«0 h m n n mm^-^m \r>^o t>. t^oo oo 
mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm 


Q 


m m -*o no^O « fi mo oo o\h m -tio c-*oo o h m t} o so\0 n m w» 


O 

s 
u 


m ^ r^ o mo on mo o> n moo m moo h -^ t^. ■* t>. O mo o> w o o>n m 
mmm-^-Th-^-'i-inm mo o o t^ t>« t^oo ooooonononOOOOmmmwn 

SSNNSSSNNNNNNNSNSSNNN t^-00 CO OO CO OO CO 0O CO 00 

mmmmmmmmmmmmmmmmmmmmmmmm<nmmmm*om 



w w n m n mcommm-<*-'*-'^- , ^-'Tfmmmm m^ 



290 



FUNCTIONS OF A ONE-DEGREE CURVE. 



£ 

£ 



Q 
< 



o 

Q 



CO N h O O OOO t»»vo »o -*■ ro N m 0>c« t^vo to "*■ "*• ro n m O OOO t-^so 1 



N(NWWWWMWNWN(NNNWNNNN(NWWMWN«NWWNe) 



^•HOO tJ-h f> "* M t>. •<*■ H N-+HCO mwoo IT)N 0\^« OvO CO O^O MO N 

t*»oo oooOOhnojcO"*-'!*- mvo vo t^oo oooOOHNNcort--.*- iovO I s * r-> 
OOOOOOOO 0>OM>0>00^(JOM>©0\OW»OnO O O O O O O O O O O O 



O "«■ O iriHVO NOO ■<*■ O vO N 00 ■**- O vO NCO "<*■ O O N 00 •<*■ O vO N 00 -^- O vO 

N CO ■<*■ -<f iO mvO vO t-*CO OOOOOw>-i«NCOTt-'<i-m i/-)*£> t^ t^OO CO On O O 
VOVOVOMDVOVO^OVOVOOVOVOVO NNNSNNNNNNNNSNN t^.00 CO 



-<t- wivo S^O h N to -<t-vo tvOO OiO h rot iovo t^CO w N CO -<f in t^oo o 



N "fr^O CO N "»hv© 00 N "*-vO CO O W ^vO CO N tJ-MD CO O <N ihvO 00 
H h m m m oi « n w n rnroforoco^'t't't'tioinioifl mvo 



© 



A 



55 



N "*vO 00 N "*vO CO N t"O00 N "*VO 00 O W ^vO CO N "<J-vO CO 
H H M M H CS N CM M CM MrOfOfOrOt't't't'tlOiniOlO lOvO 



u"> Tf- co N h O OOO t-*\o iflMNH OOO txvo uo^mn w O t^vo »r> ■<$- co 

lONO>H CO lOvO CO O N ri-vO CO O N fOlONOH CO m tv O M W "*vO CO N 
ooooooooooooOOOOHHi-iHHHNNNNNcococococo-<s-Tt- 

OOOOOOOOWHMMMHHHHHMHMHHMI-IHMHMMH 

NNNNCMWNCSN«WNM«WWWW«WN«WNWNNe)«e»W 



N-tO SfiOvO co OvO ro OvO N CMONM m h CO -tHCO * h CO iJ-hOO ■<*■ 

t>.00 OOOhmNNCO'*"*- lOMD vO h*00 OO O\0 O h N N fOt* lOO vO t^ 
vO vO vO vO t>.t^l>*t^tN.t^t^t^t->t^t^t^t^t^ t^OO oocooooocooocooooooooo 
MfOfOMnMrofOMroMromroMfornfororOfOcororofOfororOfOfom 



VO M NMCO Tt-OMOM tv COCO <<*• VO N SfOOM 



hvO NOO •tO iflH SfOO 



irjvo vo t»s t^»co co o h m n roro-^-Tj-in invo t^ fxco oooOOhmnn 
■>t*t i t i '*^ , ^"fiirii/)iniriioinioioinir)ioini/iiniri rnvo >o vo vo vO vo 
rnrofOMMrofornroMrorororofororororocorococororoncoromrofo 



io\o co o h co *>*• m t^co o h n co tj-vo t^oo o h n ■<*■ in*o t^ o h co -<t- 
o c< moo w moo m -^- r-% o rotN coo o n in o w moo m ^t^o ■<i-r > ->0 ro 

H f» N N mroro-t-t't^io ir>vO yDvflvO NN t^OO OOOOOOOO Q m m 

ooooooooooooooooooooooooooo o o o 
fOrOfOrofOfOMfOfOfOm«fOfOfOfOrororonmrofOfororO'<j-'tf*t 



FUNCTIONS OF A ONE-DEGREE CURVE. 



291 





1 

CO 




N -**0 00 N Thvo CO ON -*vO MOM -*vO 00 N ^vO CO N ThvO 00 
h m h m h <n n n N n mcorofom^'t't't'tioiflinin »nvO 


Q 
< 


H O00 t»» f^vO mifltflN N h OnOO 00 t^ t^NO mifl^MMN M M 


o on h n tj-vo oo n -<*-vo oo o n -<*-no oo on w ro m t»> on m common ro in 

vn mvO vO^OvOMD C-.f^t^t^ t^OO 0OO0000000 On On On On On O O m m m 
NNNNNNCNN<NClNC)N0)C')NNNNNN<NCSrOCOCOCOCOCOCOCO 
CM<N<N«NNNNNNNNNNNNNN(NNNNNNNNNNNNN 


u 
w 
w 

X 


vo roo t^-<j-Hoo m n onvo m o oo m n onvo -tHOO in n o n-^hoovO co 


coOnOOwnnco-^-^- m^O t^ t>.00 0\0 m n M CO t»- in invo t^ t^oo ON 


Q 

o 

Q 




00 OnOnQ h h « fOfO'*'* mvO VO t^ t"-00 OnOnO w m N N CO -<f ^ in tOMD t>. 
OnOnOnOOOOOOOOOOOOOOOOhwhmhhhhmmmh 


Q 
K 
O 

U 


H N CO ■<*• 10*0 tN.08 ON H N CO ■<*■ in lOvO t^»00 ON H N CO ■<*• mvO t^OO On 


COvO ON N lOOO H -<*-00 H -<*• t-H CONO On N inCO N lOOO H t(- N COvO On CO 
OOOOHHWNNNCOCOCO-^-Tj-^-^iniO mvO VOO NN t^OO OO OO OO ON 
NNNNN<NNNNNNNNNNNNNNNNNNNNNNN04NC-1 


2 

s 


N "^-vO WON "*-<0 00 N -<*-vO OO O « t)-nO CO N -^-vO COON -<*-vO 00 O 




2 

3 


N ^-vO COON tj-vO 00 N "^vb OO N «i-vO 00 N ^vO 00 N tJ-vO CO 
H M M M M N M N N <N COCOCOCOCO-^--*-'* , '<*--<t-inininin 10*0 


p 


lOTj-rOCON H Q\ OnOO txvO in •<*■«*- CO N H O OnOO t^vO VO m ■*■ CO CO N H 


On w romNOH w -<i-\o 00 O N ^i-vo oo N ^j-no noh romsOM miON 

OnO M H M M M N W N W N COCOCOfOCOrOTt-Tj-rj-rh^-lOlOlOXO 
mNNNNNNNNNN<NNNC-)NNN<NNNNNC1CJ<NNNC)<NN 
NNNN0)<NNN(NNNNCN1NNN<NCNNNNNNNNNNNNNN 


u 

w 


SrOO N'tHOO ION ONVO COOvO COO S'tHCO ION OnnO CO N "+ M 00 vO 


r-NOO onono h m n coco^- iono vo r<.co co ono h « n mtm tn^o t^ t^oo 

OOOOmmi-ihmihhihhhwhhi-iNNCNCNCnINNNWCNWNN 


Q 
O 
P 


VO NOO tJ-OnO N00 -*0nD NOO -*0vO NOO •^"OnO NOO -<4-0nO MOO <* ^» 


OHMNrOrO^J-'^- lOMD VO t-x t^OO ON ON O M N W COCOM-iO ionO no t^OO 00 
COOOOOOOOOOOCjOOOOOOOOOOOOOOOOOOO OnONOnONOnOnOnONOnOnOnOnOnOnON 

cococococorocococococococococococococococococococococococococo 


d 

K 

o 
a 
u 


ON H N CO -<*• ionO 00 ON h N CO tJ- mvO tN.CO On h N ■<*- vOvO t>-00 On M 


NO COnO On n moo h rl-00 m •<*■ t^ CONO On N in On n moo m tJ- t>. CO t>» Q 
O M H h M CI N (N COCOCO"<i-Tj--^-iOinin iono vo^O nn r^oo 00 00 On On O^ 

HMWHHMMHHMMMMMMMMMHMMMMHHMHMMMM 


z" 


N tJ-vO 00 N tJ-vO 00 N TfvO 00 N ^-"O 00 N -<J-vO 00 N ^vO CO 

m m m h h cn cn N cni w rococococo-^-'^--^-'>4--^-inininin mvo 





292 



FUNCTIONS OF A ONE-DEGREE CURVE. 



Q 

< 



Th-^-fOrONWWMHOOCT> OOO OOOO N t^vO vo vO muiifl^' 



■tfomN « 



m 

"# 



H OwO "*■ H OwO tJ- h OwO ^ H OVO ^" N OMflN O MOfOOM mflMOO 
w n m •<*- iri u->vo c^oo oooOMHNrOTh-<j- invo f^oo oooOwmncotj-^ 

l>.t^t>»r^l>»t^C^l>»t^t^ t^OO 000000000000000000000000 OOOOnOONO 



moo iohoo -^-o t^movo « o\mwoo -^-h swomonoo m h n^hoo in 
vo vo t^.00 ooooOHNNrom-^-in mvo t^ t^oo oooOOHWNro^-^m 



tOVO t«s t-^00 ONOOMNfOfOTj- lOvO VO t^OO 0> h N i 



I M ■«- iTivO vO t^OO 



inoo h t*- t>_ o tJ- t^ O covo o> N moo h tj- t^ O rt- t^ o mvo o m moo h -<f t^ 
OOOO O^O^O^0 O O w m h m n m w romro-t't'tminin u->vO vO^O nnn 



< 



m O00 oo t»» t^vo *0 ifl*Thforo« h h » O>oo 00 t*» l^vo vO ifl<0"* 

ms ov o (N -<*-vo oo n "^-vo oo O w tj-vo MOW mmso^H tmoNO^H 
h h hi w m N n n romrororo't'ti-'t'tininioinm mvo vo vo vo vo t>. t>. 

MNWNWNNNWNWNWCJWWWCMNCMCVINWWWWCIWMWW 






MOW Ov t^ ■**• ►■ 



OvvO fOMOO ioroo MOW ON*N Ov r-* -<J- h ^vO ■<*■ ► 



o h n n m^ifl iovo t-^oo oo o^O h h w mnt mvo vo t*«.oo o> o O w ., 
l/iLnini/iinioiommiAininio invo vovovovovovovovovovovovovo t>* ^ t>» 



N00 -<4-h SMOvO N OmomoO ■**• O t^ro OvvO fl 00 m h f^ -4- vO ro OvvO « 



H N N CO rT u-)vO t>s t^.00 O^O H N « W* mvO f^ t^OO OO H N « fOtm 

rovo o> N moo h ^-NO rovo rovo Ov w moo m •*• r^ o ro t^ mvo O N u-> 
OO0>0 O O m m m n n oj mmmrOT)-*'tiAin u->vo vo^o nnn f^oo oo 



FUNCTIONS OF A ONE-DEGREE CURVE. 



293 



« 

^ 





OJ *vO 00 N *vO 00 O N *vO OO N *vO OO 


O N *\0 00 N *0 00 

* •* -<i- ■* ■<■ io io >o m mvo 


< 


<n^*'*'***'tfnmMromrofO« N « oj <N 


NNdNNlNMMHMH 


m romNOH m^NO^H ro m t-^ o> h romNO 
OOnO>0«000000'hi-'i-(Mm(NWNMM 

NNNCNCNNCNNCNOlNNlNNlNtNNNCNN 


NNINCNNNINNINCICN 






fOOOOvO * N 000VO * <N OOOvO * N 00 vO * N 00 vO * tN OOOvO * CO 

00 OO>0 h N CO CO * "">^0 t-s t^OO OOt-wNfO-^m ir>vO t^OO 0> 0> f IN 
m m !-■ n n cm cm 0) w cm cm cm cm cm cm mmmfOmmmronmfororO't't't 



Q 

o 

Q 


cm om«oo m«oo io h nmO smo sroo N'+hoo m cm ovo too n^ 


ir> \no t^ t--»oo O <-• m cm r<iro*m iovo t^ t^oo OOO m « cm co * -*j- io 
NtsNNtststs i>.oo oooooooooooooooooooooooooooo oonOOOOnOO 
******************************* 


Q 


t^ t^OO 00 U»0 H H N N MfO^tm IDVO vO tv t^OO OOOOOOhmCM 


K 

O 
X 

u 


o cm iooo ii -<i-oo h- * t^ o covo o cm inoo i-i * f^ comd o>« i^O\« moo m 
vo nn r^oo oooo O O hi m hi cm cm <N mnnmtt'tinm ino 
iOi/)iomi/lirii/iioir) invO vO v OvQvOvOvOvO , OvOvOOvOvOvOvOvO v O v O , OvO 

******************************* 


fc 

s 


CM *vO 00 CM *vO WON *0 00 CM *vO 00 <N *vO 00 CM *0 00 O 

m m h m h cm cm oi cm 0) cococococo*****ioiomin mvo 



o 
V 



MMHMHCMCMCMCMCMl 



CMCMHmhhiOOOOOO O00 00 00 00 t^ t^ t^ f^vO vO vO vO vO iflWiniflifl 
CM *vO OO N *0 00 Oh COIONOih m^NOvH m lO M> H CO ui N O H 

tomrom't't't't-tti'iiomin invo vOvoovO t^ t^ n r-» t^oo oooococo o> 

***************************-*-TJ-T*-^- 

CMNCMCMCMCMCMNCMCMCMCMCMCMCMCMCMINCMCMCMCMCMCMCMCMCMCMCMCMtN 



[ oo>o m cm o>momocovO * h o t^ io co o oo vO * h osmmH osinro 

I * u-)\0 t*. t^00 OiO h H N «** mvO t-^00 00 OOhhiCMCO*iO u">vO t^oo 
OOOOOOO0000000000000hihihihihihihimhihim 

^■^'t't^^'tioioifliomiominioininmmioioioioioioininininin 



iOmOO * hi N'tO NfOO SfOOvO rOOO CO OO CO OwO CM OvO CM O *n OJ 



MOM>0 m« M M** lOvO vO t*» txOO OOvO h h N N fO** lOvO vO t^ 

NO fONO rOvO 0> N iDOO m -^ N fOvO OO^O O N U100 hi * t>» (OvO On 
t»N0O 0000 00000 O hi hi hi n n tN IN rorom***ir)in idvo «o vO vO 

**'t't't^ i *^ , i/iioiomi/ , )ioi/iinioioininiriifliriir)inioinini/imi/i 



m 



FUNCTIONS OF A ONE-DEGREE CURVE, 






l M H H N N N N <N rOfOfOrOfO't-t't't'+u 



Q 

4 



MfororofJ^i-^^'ti-ioinininin invo vo^o^o t^ t^ t>» tvoo oooooo OiO» 

m ro in t^ on w roiotvOH muiNaH rtWNOH co in t^ o< h en 10 t^ o> w 
h h h h w n n n n n roMfororO'+'fti-'+inininio invo vovOvO^o s 
\pvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvO 
WNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN 






w 

Q 

o 

Q 



000 MOrOO 000 MflfON O. t^vO rhfOH 000 fx in -<*- N H OnOO vO in ■>*■ 

r>. t-»co o> h n n m **• invo t^ t^oo o m n ront invo r*>co oo o> h n 
vo vo vo vo t>.t^t^.t^t^t^t^t^t^r-,i>. t^oo oooooooooooooooooooooo ooai 
iouivou , iiflif)ini/)ioir)i/)inininioioir)inioioioioioioiflioioioioioi/) 



OMT> M dWO COO t^Tj-H00 in N OwO COO N * H 00 ION OnvO CO ts ■+ H 00 

lOvO tx t^OO OnOOhNNco-<*-->*- invo f^ C^OO 0> On O h h N cOtJ-^- invO vO 
HHMHHMNNNNNNNNNNNNNNNCOCOCOCOCOCOCOCOCOCO 

ininicioiflviioioioioioiflioinioioioioioioioioi/iiniriirii/iiflioiolo 






2; 



fO ro ■* •+ -<j- -t m »fl invO VOVO NNS t^00 00000nO>0n0000hhhNN 

N IOO0 H iJ-NO COvO On N IO00 H -tNO COM3 O^N moid m00 H i-NO fO 
lO ID invO vflvO NSN tv.00 00C0O\O\O>000OHHHNN«fOnfO + 'J- 

t^tN.t^.t^t^t^t->i>»t^.t^r^t>.t^r>.r^ t-^oo cooooooooooooooooooooooooooo 

•^•■'♦•■<*-'^-^-'<*--^-'^-'<t--^--*-<i-'>4-T*-^--<i-Tj--^-^-Tf-^ <*-Tj-Tl--<t-r}-^--^-^-<j-rJ- 



Q 



HHHHHHHHHHHHHHHHHHNNNNNNNNNNNCOCO 

m roiosOH roioNO>H mmNOH mmNo^H romNt^H mionoh 
in io m u~) invo vovovo-o t^r^r^t^ r>.oo oooooooo o> on o> on on o O O O w 
inininmininininininininininininininininioininin invo vo vo vo vo vo 

NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN 



X 

^ 



coh ONts.in^N OoovO lOroH on t-^vo -* a ooo Mnro« 000 MorON 

« fOfO't mvo t^oo ooo^O h n n m* mvo t-» i>»oo on o h n n co <<t- invo t^ 
Tf-<4-Tt- , ^-'<j-M-'T}-T*-Tj-Ti-inininininininininioin invo vovOvOvovovovovo 
ininininininioinininininininininioinioininininminininmioinu-) 



"*- S'tHOO -^-HOO in N 00 ION 0>vO N OnvO CO VO COO N "t H 00 ION ON 
invO vO t^OO OOONOOHNNCOTj-Tt- invO vO t^OO ON On m H N ro m •* m m 

ooonoonononooooooooooooooomhmhhmhmh 
■^■•^■■^■•^■•^■^^■loinioinininininininininininminininioioioioinin 



N N CO CO CO •«*• -<i- in in mvO VO^O NN tvOO OOOnOnOnOOOhhNNNCOCO 

w ■**• r^ o comd on n moo h ^no covo on n moo m moo m -<t- r^ o covo o> n 
vO VO VO c^ r^ t^ i-^oo 0000 On 0> ON O O O O M m h N N N cococo-<4-Tt--<i--^-m 
vovovovovovovovovovovovovo t^f^c-»t^t^.t^t^t^t~Ni^srvt^r^t^t^r^r^t^ 



Z 

s 



FUNCTIONS OF A ONE-DEGREE CURVE, 



295 



55 

3 


CM ""^^O 00 


CM "*-vO 00 CM 

H M HI M M CM CM 


^"O CO O N -^""O 00 CM "xhvO CO CM -^-vO OO 

cm cm « roconforO't't-t'<j--tioioioio mvo 




H M CM CM 


CO CO rt- tJ- in inO 


r^ t>»oo ooooOOMCMCMcoro^'* invo vO t*» 


mmtsOH 
ro ro ro ro ■>*- 
t^ t-^ c^ t-^ t-« 

CM CM CM CM CM 


nioNO^H m in 
^- rj- -t Tj-m m in 
t^ t^ r^ t^ t^ t^ t^ 

CM CM CM CM N CM CM 


t^Ow romNO CM -xJ-vO CO O CM tJ-vO 00 N * 
LO IOVO vOOvO t^ t^ t-» t*» 1^00 OO 00 00 00 & & 0\ 

CMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCM 




in ro CM m 


o t^vo m^-fOM 


o ooo t^»vo in co cm h o ooo t^o m'+mt) 


C/3 

X* 

W 


oooOi-iCMCMm-tf- ino t-^oo o O w cm m ->f invo t-voo oooOMCMro^in 
m m cm cm cm cm cm cm cm cm in in cm mmromrommmmmmm****^^ 


Q 
K 

o 

Q 


CM O t^ •<*- M 


oo u-> ro t-^ ■«*• cm 


OvO MhCO ION kf."<J-Ht Qs\Q CO 00 in N O 


OOOOOOi-'i-'CMcO'tf-^- invO v£) t-v00 O O O hi CM CM m tJ- -^ invO t^ 1^00 O O 
m in invO vDvOvOMDvOvOOvOvOvOvOvOO t^r^t^t^i>»t*»r^r^r^r^t>»t^t^t>» 



ro\0 O n moo m Th r^ o rovo o cm moo m ^-so rovo o cm moo m rt- t^ co : 
f)cnroi--t^-iriiri in^O vO^OvO SN t^00 0000OOOO000mmhiCMCM | 
OOOOOQnOOOOOO o-o> OOOOOOOOOOOOOOOOO i 
^^^^^■«--«--^-Ti--^-^-Tt--«-^-Tj--<*-^-^--^-^-«i--<}-^inininininininin! 



U 



OOOOOhhhcmcmcococotJ-- 



- in \r>\0 ^OvO N t^00 00 00 O O 



it COvO 00 CM tJ-vO 00 O CM ^-\0 00 O CM Tf-vO 00 CM tj-vO OO CM tJ-vO O m CO 

i^ r>. r^ t^oo ooooooooooooooOOOOi-iHHiMMCMCMCMCMCMroro 

CMMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCM 



I 0\00 t>» lO tJ- ro I 



O t>xvO in ro cm 



CM ro Tf -*• invO t-^oo o C 
CTiCJ>00><^OvCT>ONO>C 

loiflinmiomiom in^c 



m cm ro -t- invo r>. r-oo 
- O O 



ooo f^ m ■<*■ fO CM H ooo t^vO "*) 
N ro*t u")\0 t^OO 
vOvOvOvOviD'OvOvOvO'O 



5 t>. f-00 On C 

) O O O O i- 
5 vO vO vO vO VC 



00 in CM On^O WO NtH OvO COO Si-HOO mN S^hOO ION OivO ■* CM 

vo r>«oo co c>0 h h n nn + iovo "O f^oo oooOwHCMmrO'^ir) vnvo t^oo 
fOfororom't'+-t^'t-!)-'t-t'fti--t^ij-miomininininmioininm 
loininioioioioioioiommininwiioinminmmioioioioioiomioioio 



cm cm cm cm cm rorofororo(T)fO^-^-^Ti-^--^--^-rhiri»riir>ioioin mvo vO vO vO 

rovo O cm moo h t)-no rn^o o cm moo m -^ t^ rovO 0> cm moo h tso ro 
* ■+■<*■ mm mvo vo« nnn r^oo oooooooOOOOHMMCMCMCMcoro 

OOOOOOCOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOO 



N "*V0 00 CM t*-vO COON ThvO 00 CM tJ-vO CO N *vO 00 N '♦■vO CO 

h h m h m cn cm n cm cm roroforoco^*t**inu>inio mvo 



296 



FUNCTIONS OF A ONE-DEGREE CURVE. 






10 



x 
O 



2 






u 






On On h CM ro * io mvO t^OO On h h cm ro rf- m\o t>.00 On h CM M^irjifl 

vO 00 h ro m h* On h roiONO>H rj-vo 00 O CM rhvo O0 CM i-NOH MiflSO> 
m mvo vo vo vo vo t^t^t^t^ i-%oo oocooooNONONONONOOOOOHHHHHt 
oooooooooooooocooocooooooooooooooooooooooo & ao o oi o\ c> a OMJ 

CMCMCMCMCMCMCMCMCMCMCMCMCMNCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCM 



t*» t^vo lOTt-^fOCMCMHOOON OnOO t^ t>.vO vO mm^-'J-rOMN h h O^ 

« ro* iovo tvoo On O h cm co ro ■**• mvo t-^oo uo h « rn + mvO t^oo On O 
t^» t^ t^ t^ t^ t^ t^ t^OO OOO00OOOOOOOO0OO0OOO ON Oi Oi Oi O (> OMJi O (J 
vOvOvOvOvOvOvOvOnOvO'OvOvOvOvO'OvOnOvOnO^O'O'OvO'OvOvOvOvO l>it^ 



Ooo toroooo tomooo \r>roOoo mroOoo tnroooo iomhoo m ro oo vO 

n n m4<o iovd r^oo co 0\0 h h « ro44 »nvo tx t^oo ONOOHCMmro-* 
OQOOOOOOOOOmhmhhwmmmhmmmnmcmnnNN 

vOnOvCOvOvOvOvOvOvOvO^OnOvOvOvOvOvOvOvOvOvOvOvOvOnOvOnOnOvOvO 



in m ^ ■* ^ ■* ro n fo ro « cm cm 



imhOOOOOnOnOn OnOO 0000 NNb 



iomioinininioioiomioinininuNioioio>oiriioinioioioiDinio»oini/N 



i h cm cm cm cs cm rororofoco-^-'^--' 



I H CM CM CM CM CM COCOrOfOCO'*'^-'^-'^"'' 



txOO 00 OiO H N « M'ti- u"NvO vO t*»00 CO O\0 H M « m'tm irjvO t^OO On 

»*-vO 00 O fOmtsONH M»Ots©H MunNOnh Tf-vo CO O CM tl-vO 00 O CM -«J-vO 
OnonOno O O m h h h h cm cm cm cm cm mrofom*'*'*'tT)-winioin 
t^ r^ t^oo oooooooooooooooooooooooooooooooooooocooooooooooocooooo 

CMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMeM 



CM h On OnCO t^vO lO «$■ CO CM CM H O>00 t^NO U") U"> -<*■ CO CM CM H O On OnOO t^ 
lOvO t>» t^-00 On H CM CO tJ- mvO tvOO 00 0*0 h « «•* \T)\0 t^OO ON On H CM 

^■<j , *'*'*'+»n»ominuN.iom»oinm mvo vovovO\0\o^ovovO\Ovo t^t^t^ 
vOvOvOvO v OnOvO v OnOvOvOvOvOnOnOnOnOnO , O v OnOvO , >C , >OvOnO , , OnOvOvO 



Onvo rot-toomroo t^. »o cm On^tj-h onvO mhoo loroo t^ in cm o> t-* -"J- CM 
o\0 h n « m-tuN invo (^ t>»oo o\0 O h « mm* mvo vo t-*oo oo o> Q m cm 

t^OO OOOOOOCOOOOOOOOOOOOOOOOO OnONOnOnOnOnOnOnOnOnOnOnOnOnO O Q 

ininininiflioinininininmiomioininiomioiomininininuN tovo vo vo 



t^ts.t^t^t^t^t^lNits. txVO vOvOnOvOvOnOvOvOvOvOnOvOvOnOvOvOnOvOvO m 
COVO On CM lOOO H "«J- t*» fOvO O CM tDOO H * t^ O fOvO On CM ""NOO H tSO « 

cm cm cm cococoTt-^^mmm m\o >ovo ns t^co cooooo on on on o m m 
OOOOOOOOOOOOOOOOOOOOOOOOOOhmmmm 
ioininio<ninuNininioiommioioioininiflioioioioio>oiouNin»n»o«n 



i m cm cm cm cm cm commmfO't^'t + ^ioiomio m\£ 






FUNCTIONS OF A ONE-DEGREE CURVE. 



297 



i CM cm CM CM CM cococococOTf-<*--<i--«--«t-ini/-N»j 



< 



oo oh cm co Tt- in soo On cm co "*■ »o ts.00 on h cm co mvo soo h cm •>*• »ovo 
cm -4- s on h ro m s o> h ^vo oo o cm -^-vo co h mion^h ro\o oo o cm -«*-vo 

COCOCOCO Oi^O\0 0\0 O O O M H w H H CM CM CM CM CM fOmrOfTt't^Tj- 

OnOnOnOnOnOnOnOnOnOOOOOOOOOOOOOOOOOOOOOO 
cm cm CM cm cm cm cm CM cm cocococococococococococococococococococococo 



<y> on o o«oo cocococo sssssssssssss svo <o vo vo \o vo <o vo 

On O f CM CO ■*■ invO S00 0>0 h N M+ mvO SOo" O m CM CO ■*- lOvO S00 O 

cm rocococococococococo-^-"^-"^Tj-Tj--^^->4-Tt-^-ir)inij-5u-)inir)inu-)toio 



o 

Q 



-* H O^O ■<*• CM S lO CO H 00 VO -tf- CM 0> S lO CO 00 VO "*" H OiMOrOH 0*N 

S00 COOOMMNcOThio iovo soo cooOhnnco^io iovO soo o> o 



2 

s 



SVO vO lO m "*■<*- CO CO CM CM l 



l OnCO CO S SvO v© U"> tO -<*• tJ- CO CO CM CM H 



h *so covo o> cm moo h tso ro moo h <^- s o covo On cm *ooo h ■«*■ s 
On o> On O O O O m m h cm (N cm cocococo-<4-^-"<*-ininio mvo \OvO nn soo 
CM CM CM rocococococococococococococococococococococococococococo 



< 



tOvO CO O* H CM CO "<*■ 10"0 S00 On H CM CO lOvO S00 0> H CM M- lOvO S00 

On H fO lOCO O CM rfvO CO CM -^-vO Onh m W SO h CO lOCO O CM -^-vO 00 CM 
h cm cm cs cm rorOifOrom-t^^-+'tinwNinm invo vo vO vo s s s s soo co 
OnO*OnOnOnOnO>OnOnOnOnOnOnOnOnOnOnOnOnOnO>OnOnOnOnO>OnOnOnO>On 
CMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCM 



Q 
as 

o 



On OnCO 00 00 SN SvO vO lfl «fl ifl -t •* M fO CO W « CM H h i 



h On 



O i-t CM CO •<*• lOvO SCO O* M CM CO "*• IOVO SO0 On m CM CO -<*■ lOvO SCO On On 
OOOOOOOOOOmhhhhhhhmhcmcmcmcmcmcmcmcmcmcmcm 



VO ■* H OnnO ■«- CM On S ■«- CM S li"> CM 000 lOCOOOOV© CO H 00 VO tJ-h OnnO ■* 

■*■ IOM3 vO S00 OnOnO h N COCO-* idvO vO S00 OnOnO w M N CO ■* in lOvO S 
CM N CM CM <N CM CM CM COCOCOCOCOCOCOCOCOCOCOCOCOTj--«i--^-Tt--<1-rh-^-Tl-Tt--^- 

vOnO'O'O^OnOvO'OnOnOnOnOvOnOvOvOnOnOvOvOvOvOnOnOnOnOnOvOnOvOnO 



S SvO VOVO lOlOXO-t-rhThrOCOCOCM CM CM H H I 



ON ON ON00 00 00 SN 



CM tOOO m ■<*■ S O COVO ON CM IO00 H +N0 COVO ON CM IO00 COvO On CM tO00 H 
OOOHHHWNNNfororo'+'t'tioiflm lovo >o >o S S S S00 00 00 o> 

CMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCM 

ioinmuNioiomifliniflininmininin>ninioinioininini/Ninioininioio 



i h m w n « « « focorocoroi)-^ 



FUNCTIONS OF A SnEDEGREE CURVE. 



10 



I CO *vO SO»HN *vO t^ On N 'ttntsO^O N ^"O NO\H M *vO 00 h 



cocococococorococococococococococococococococococococococococo 



w cj n cocococo*Tt-*io mvo vo t^ t>oo oooo ao»0 o h h n n mm** 
O h « n* mvo t-»oo C\Q m « mt mvo f>oo (>o n m* mvo t-^oo o> h 

0>0nO0n0nO(7>0nO0>0000000000hhmhhhhihhNN 

t>t>NNtstNt>Nts t^oo oooooooooooooooooooooooooooooooooooooooo 



* N OOOvO * N OOOvO * N OOOvO * N 00 vO *N 000VO * N OOOvO * 
* invO vO t^.0O ?5 5 H N fOJJ ipvO £»00 CgONOHNNCO* mvo VO t^.00 



N m O O 0*00 t^ t^vo in r»- CO CO N h OnOO t*s r^vO m * * CO N h O On 
00 H * t*» ON N u")O0 H * tVQ COVO 0*N WNO COVO ON lOOO M * t^» CO m 

\0 nnn t>.oo oooo o> On o> O O O m m m n n n m ro m m * * ^- m io m 
***********inu->ir>u-)ir)inioir>ininu-)ininioir)ioir)inioiO 
inininininininmioininioinmioinioirnnioinininiflinioiomioinir) 



i h h n n w « vn cocorororo****'«i-ir>»oio»o mvc 



<0 
»0 



o 



fc 

S 



« *vO 00 N *vO OOOO *vO 00 N "*-vO 00 N *vO WON *vO 00 
H h H H M M N N N N roMrococo**"t*"*io»nioio lOVO 



VO 00 ON H N * U">nO 00 On H « * mvO 00 ON H N *VO N»0 N W lOvO 00 On h 

VO 00 fOiONONH CO moo 6 N *vO 00 CO \r> t^ ON M COVO 00 N *vO 00 M 
Th^iOlOiniO mvO VO NO NO SSNS t^OO 00000000 On ON On ON O h 
OOOOOOOOOOOOOOOOOOOOOOOOOmmhhhm 

COCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCO 



vOvOVO NNNNNNS txOO 00000000000000 OnOnOnOnO H l 



t^*CNl ON tx * CV1 OOOVO rf N OOOvO * N OOOvO ""*■ N OOOvO * N OOOVO * 

O H « n ro* tovo VO t^OO OnOOmMCO** iovO f^OO OOOO M N N m-t 
NtstsNNNNNtNNN l>.00 0000O00000000000000000O0 On On On On On On 
nO no vO vO vO vO vO vO vO NO vO nO vO vO vO vo vO vO vO vO vO vOvovOvOvOvOvOvOvOvO 



m o onoo oo t^ txvo mm^roMN h h o onoo oo t^vo vo io in * co co (s 

0* COVO 00 H -tNO COVO On N IO00 M 4 N O A »T)00 M «4" tx 6 COVO ON N tf)00 
oooooooo On On On O O O h h h cm N N cocococO*rf'<l-»r>inir) invo vo vo 
fococororocoro***********'************* 
vNinwiintf>inininir>»oir)ininioio>oio>oir)io»nini/Nir)inio»oioin>om 



MMHMMWNNNWCOCOCOCOCO^ 



FUNCTIONS OF A ONE-DEGREE CURVE. 



299 



O CM *VO 00 O CM *vO 00 O CM *vO 00 CM *NO 00 O CM *NO 00 O N *VO 00 O 

h m m h h cm cm cm 04 cm rororororo*****ininioin invo 



iSO»H fOiONO^H romNOH *nO 00 « iflM^H fO in00 CM 

Q h tJ-^o oo d cm in t-^ on h ro\o oo d cm * t^» on m rovd oo d cm 4- t^. on m ro«o oo 
* + ■* t ui 10 m m mvo vO\0\o t^ t^ t^ t^ t>>oo oooooo o\om>o<^0 o 

CMNNNCMNCMMNCMrNrNNNCMrNCMCMNCMrNJCMNCMe'JCMCMrOrOtOrO 

<i fOfnrOfOfororomfOfnrOfOfOfOtnrnrocororororornrorocofOMrorom 



invO t^OO ©OO m« Mt IDVO t^OO O»0 m « M* invO t^OO O\0 h N to* 

CO * mvO t^ On h CM ro * invO t^OO 0> H M tOt invO t>»00 0> O CM ro * lOvO 
in m in in in m^O MD^OvOvOvOvOvOvOvO t^f^txt^t^t^t*«.t>» t^CO OO 00 00 00 00 

oooooooooocooooooocooocooooooooooooocooooooooooooooooooooooooo 



2 

2 



On f^vO * CM i 



ON t*% in * CM On t>. in * CM On t^ in * CM 000 t^ 



n m-t mvo vo t^oo on m h cm ro * m^o <o t^oo On h h cm ro * m\0 vO t>* 



OnOO t^vO in + MN 



l OnOO ^ in * CO CM m o Onoo two in * ro CM H 



rovo oo h >tso rovo o\« mso ro^o On cm moo m rovo on cm moo h *so 
•>t rt ~4- ir> \r> invO O^OVO NN r^OO OOOOOOOnOnOnOOOOmmhNNWCO 
vOnOnOvOvOvOnOvOnOnOnOnOnOnOnOnOvOvOvOvO SNNNtNtsNNNNN 

lninviioioioiflUNioifliflifliflioioiniflinioioinioioinininioioinioin 



MHMHHCMCMCMCMCMrororococo*'< 



< 



h ro to tx ON M CM *vO 00 CM "tiflNO>H fOi^NONH «iflNO\H CO in t^ ON 

vooo cm 't nonm ro inoo Q cm *nO oo m rointsoo *no oo romNOMH 
t^ t^.oo oooooooooNONONONOOOOOMHiHMi-iCMCMCMCMrororororo* 

HWMMWMHMMMMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCM 

fOfomrocnfororomfOroMrOfOromrorOforocorofNfOromfncofOfOfO 



* in in<0 t^ t^OO ON On h h CM CO CO * * invO VO txOO OnO h CM CO * in in 

m CM ro * invO tvOO Oh fl ro* invO t^.c© On h CM ro *vO t-^00 (JO m N to 
cm cm cm cm cm cm cm cm cs roMfOrorofOfOfOnTt'*"<J"*-'t'*'<t-+Tnn»nuNUN 
00000000000000000000000000000000000000000000000000000000000000 



-CM OOOvO * CM OOOvO lAfOH 0> Mfl fO H On two * CM OOOvO * CM OnOO 

CO OO O h N to** lONO t-xOO OO O\0 H « n ro* mvo t» t^OO ON H w N 
m m cm cm cm (N cm CM cm cm CM cm cm cm cm rorororororororororororo**** 
t^t^c^t>.c>.t^c^tvt^t^t^t^txt^t>.t^t^t>Nt«*txt>.c>.tN.tN.t^t^t^c^txt^rs 






Onoo t>vo m * ro cm cm h o Onoo txvo in * ro cm h o ooo oo tvvo in * ro cm m 

inco h * r-% o rovo on cm in t>. rovo o> cm moo h *vo On cm moo m * t>. o ro 
m mvo ^ovO nns r^oo oooooNONONONOOOMMMMCMCMCMrororo** 
inmmmminmmmmmmmmm mvo vovovovonovOnOnOvOnovovonovo 
mmmmmmmmmmmmmmmminmmmmmmmmmmmmmm 



s 



m m m m m cm cm cm cm w rorocororo****'^-mmmm mv£ 



300 



FUNCTIONS OF A ONE-DEGREE CURVE. 



W Tj-t^ON «*■ t*. 0> W •<$• t^ O* N -^-t^ON lflNO rO lOOO H rovO Om * N 0> 
10 t^ o h 'd-vO oo ro m t-N o\ a -i-^o oc i m ro 1000 O n -tso\H rovo oo 6 w 

t^ t>» f^OO 000000a>0 > >0^OiO>0000HMMHNNNNNr0r0r0r0"^-'<*- 

focororoforoforofoforomrofocofocororoforororoforomromcorofo 



N ^ mvO 00 On H N CO mvO 00 0> h ro -<J->0 t*«. 0> h N •<*• in t^oO W CO m<0 00 

O h n ro Th m t^oo o> h w ro mvo t^oo o> O N ro ■>*■ mvO t^ o» h n ro 4- 
NN N N N N w N N rororororororororOTj-'*--tf-'<i-'<*-Tt--tf-Tj-ininininm 

0^0^0^0^0^0<O^O^O^O^O^O^O^O^O^O^ONO^O^O^O^O^O^O^O^C^O^OlO^O^O» 



Ox t^vO "J-fOH ooo r^ in •<*■ N l 



O>00 vO in ro w 0> fvvo Tj-rOH Ooo (>. in ■<*■ 



N ro Th tovo f*>oo ooo>OHNrorO'<4- mvo t^oo o> 0« m n ro -<*• m mvo t-^oo 
OnO>0>0.0>O^OnO>0>OOOOOOOOOOOOhhi-immi-.hmhm 

t^t^c>.t^t^t^t^t^ t^oo oooooooooooooooooooooooooooooooooooooooooo 



■!*■ ro h o 0> t^vo m ro N h o>oo t>» in -^ ro h 0> t^vo in ro n h onoo t-s in «*■ 

vO 0» N inNO rovd a« mNO rovO 0> N ino6 rovO Oi N moo 6 rovO 0> N 
h m cn oj w mromrO't't'tininin mvo vovO nnn t>.oo oo oo o> o> o o> O 
oooooooooooocooooooooooooooooooooooooooooooooooooooooooooooo o« 



O N "*-^0 00 N "*vO WON "*vO 00 N "*-VO 00 N ^vO 00 N -<J-vO 00 



(4 TfNONH rovO 00 roifltsOvN -»*-vQ 0» h rovO oo ro inoo N to t^ O N 

oo O N tJ-no*h rovo oo O « -t NOiH rovo oo N msOH tj-vo oo ro io 
O h h m m h n w « N rororororO'"*-'^--4-^-ioiou-)io invo vo vo vo t>» t^ r^ 
rororororororororororororororororororororororororororororororo 
rororororororororororororororororororororororororororororororo 



■^- in t^oo OO h toi- invo oo o* h n ■«*■ invo t^oo h ro rj- in t-xoo Ohm 

vo t^oo 0> N ro «*• mvo t^oo »h n rot invo f^oo h n ro ■<*■ io^o oo On 
000000000«OiOO>0>0>0>0>0>OOOOOOOOmhwhihhmihmCN 

oooooooooooooooooooooooooo o\&o^o y >o y >o\<J\o\0\0\<?><?'<yi<j>&iO > '<?<<y> 



, t^ in ■*■ N h o> t^vO ^«h o»co <0 inroH Ooo M^roo o Oit^io-" 



t^c^t^c^t^c^t^t^t^i>.t>»t>»c>.c^t^t>.r 



r^t^r^t^t^t>»t^t^t^t^t>.t»t^t>. 



s 




Q 


O«00 vO UT+rON O>00 t^vO "«J- ro N m o oo t^vo »o ■<*- N h ON00 ^O iflt 


O 
S 


O N IO00 h tNO fO iooo h tNO rovo r> h *NO rovO 0> N ■* t^ rovO 
rorororo-^'^-'^-ioioio m>0 ^^O st\N r^.oo oooooOnOiOiOOOmmm 
f^t^f^t^f^t^f^t^r^t^r^f^f>»t^t>.t^t^t^r^r-^r^f^t^t^ f^oo oo oo oo oo oo 



N '^■nO 00 W "^"VO 00 O n "^^O 00 o n -*vo 00 n "*-vO 00 o n "*-vO 00 Q 
MMMMHWNNW<srororororO'<-'^-^-'^-^-»om«oio u->\5 



FUNCTIONS OF A ONE-DEGREE CURVE. 



301 



2 


N *tO 00 CM "*-nO 00 O CM *nO 00 N ^vO WON >«-vO 00 N *vO 00 
m h h h h cm w n cm cm rocorocorO"<* > '<*-''*"«-''iJ-»ninio»n »o\o 


< 


fOvo O CM moo h ij-SO conO O CM IO00 h *SO conO COvO On CM lOOO m ■«*■ 


m ro iooo O cm insi^M -*-vo oo h to iooo O cm insa« -^-vO oo h n moo 
M m m h cm cm n cm cm ro m ro ro •* ■* * * m ui m io u->m3 vo >0 vo r^ t^ f> t>»oo 
loinioioininiotnininmininiomininioininiflini/iioifliniomioinin 
corococorocororocororococococorocorororocorocorococococorororo 


u 

w 


romsOH romtNC^H romso>H rj-vo CO CM vinom covO 00 n ^ In 


CO 

W 


O ii n ro iomD C^OO Oh cm ro »*• io CnOO O CM ro •<*■ iovO CO O m ro •<*• iovO 
OOOOOOOOOOOOOOOOQHtHMMi-ii-tMt-iCMCMCMCMCMCM 
OOOOOOOOO0000000000000000000000 



t»- ro cm h ooo svo •<*• ro cm h ooo t^vo *roN h ooo t^vo ■*■ ro N M O00 tN 

"+• iovO tN t^oo o h cm ro •<*- ■<*■ "">vO c^oo o m m cm ro rf iovo t^oo oo o 
^■^'t't't'+'tminwuominiommin iovo vonovovononononovono>o t^ 
oooooooooooooocooooooooooocooooooooooooooooooooooooooooooooooo 



COvO lOrOM 000 MOrON 000 MOCO« O tN in tJ- a On t^vO ■**■ N m o« 

tN o fovo 0\N *NO rovo Oh ^so rovo CO h -tso ro moo h -h- t^ o N 
00 OnOOOnO O O m h w m cm cm cj rorororOTh'Hr-rtrioioto iono vOvO NN 
OOOOOOOOOOOOOOOOOOOOOOOO OOOOOOO 
vo iO to to iovO vOnovOnOvOnOnOnOvOnOnOnOnOnOnOvOvOvOvOnOvOnOnOnOnO 



ON msO COvO 00 m ■<*■ ts. O CM UONO COvO O N W) N COvO O CM IO00 CO 
N U0tNO>N "*-vo 00 t-i rouiNO CM ^-NO\m rovo OO ro in t^ o CM -4-vO O M 

^■Tf'ti-minio iovo no >o no t^ t^ t^ t^ r>»oo oocooo oooooo O O m 
coroeorococorororororororororororororororororororororororororo 



00 CM romNCTiO CM "^"VO 00 O H CO IONO 00 CM tJ-vO 00 CM Tt-U1N(J\H CO 

rj-vo t^oo OO h ro* mvo t^oo m cm ro "1-vO t^oo O cm ro ■*■ mvo f^ O 
lOiniAiO lOvO vOvO^OvOvOvOvO StsNSNtstsS C^OO 00 00 00 CO 00 00 CO O 
OOO^OOOOOOOOOONOO\0>OiO\00>OnO>OnOnO<0«0>0>0>0<Oi 



■trOH OCO t^vO ■*• CO m O t^vO *fON O InnO in tJ- CO w O t>»vO 10 ■<*- 

OOOOMHCJrOTh invO tN r-vOO On O t-i CM ro CO * IOVO f^OO On 6 m CM CO 4- 
m m cm cm cm cm cm cm cm cm cm <N cm cm rororororororororororo*'>J--t*'+'t 
oooooooocooocooocoooooooooooooooooooooooooeooooooooooooooooooo 



•^- N h ON00 vO to CO CM On tvvO *fOH OOO vO in CO CM On t^vO rh CO w 00 

CM IO00 COvO On CM IO00 O COvO O N UO N rONO ON UltsO rONO ON WN 
OOOHHHHNCINtOrOrOrOtt^iOlOiO IOVO VOvO NNN t^OO 00 00 

ioiominmuNin>nioinioinminmuou5mioin>niominmiominmioifl 



i cm ro ro ro ro ro ^ 



802 



FUNCTIONS OF A ONE-DEGREE CURVE. 



N **vO00 O « -*vOCO O CM <*vO00 O « ^J-vO 00 O CM *-vO 00 CM **-vo 00 O 
M M M M H CM CM CM CM CM fOrOCOrOCO'^-^'^»-'^-'*ir>m»OlO ION© 



»4-00 H mOO CM in On « VO On N vO On COVO "*■ t*» H '"J-OO CM in On CON© O CO t^ M 

O cm ins<^N "3>VO ON H COVO 00 O fO inoo o cm in t^ on cm -^-no On m *$-no 00 M 
mmiOiO invO VOVOVO t^ t>» t^ t-*00 COOOOOOnOnOnOnOnOOOOmmhwCM 
nOvOvOvOvOvOvOvOvOvOvOnOvOvOVOvOvOvOvOvOVOVO NtstsNNSNNN 

rtforoMfonrorornrofOfOrororOfOrororOfOfOfOforororommmMfO 



u 

W 


O CM in Co CM »0 t*» NVO00 H -«*-^0 CM IO00 M "<J-vO On CM lD00 COvO ON N 

4" invO tsOO h « ■+ lOvO t*» ON O h CO tJ- iovO 00 On h CO tJ- in tvOO On CM 
NO vO VO vO vO NSInNNNN f*»00 CO CO 00 00 CO CO 00 OOnOnOnOnOnOnOnO O 
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOmm 


W 





CO CM H ONCO t**.vO lO'tfON H ONOO svo <fl * ro N M OnOO t^vo in •**■ CO 



t>*oo ON 
000000 On On On On ON 



m CM CO ■*• invO t*»00 On On 

-ooooooooo 

» On On On On 



n ro^ invo t-*oo coao h n fi* 

MMHIHMMHHMMMNCSNWN 

OnOnOnOnOnOnOnOnOnOnOnOnOnOnOnON 



mrOH on t^. in co h on t>. in co h 0\MO«h o co vo ■**- cm OoovO *n Ooo vo 

t>. O co moo h ij-n^n moo m covo o> cm inoo O covo on cm rt- t»» o covo oo h 
mvo vo vo vo t»» t>. t>» t>.oo oooooiOi^o\OOOHHHM«««mmnfO't 

MHWMMKHMMMHHHMMMCMCMCMCMCMCMCMCMeMCMCMCMCMCMCM 
VO VO VO VO VO NO VO NO NO VO NO NO NO VO NO NO VO VO VO VO VO VO VO VO VO VO VO VO VO VO NO 



i m h m cm cm « cm cm roforororot'*^*'tioioio«ni 



i cm cm cm cm cm cororocoro'<*- , <*-'«-'«-'«-in»n»n»n mvc 



■^oo h •*• t^ h -**-t^H ij-t^.0 Thr^O co t«» co t^ "*• r*. +nh ** r^ m **■ 

cm in t^ on cm ri-vo on h covo oo O romso cm <tNO.H -^-vo oo h m inoo O 

COOOOOCOOO OnONOnOnO O O m h H w CM CM CM CM CM rorOrOCOTj-Tf-^Tj-i/") 

irjininminininin mvo vonovonovovovovovovonovovovovovovovOvovovo 
fOMcOfOrocococorocococococororOfOcorocOfOrommrocococofOfOM 



u 

w 

X 


t^ ON CM -tNOtH -*-NO 0\ H COvO 00 h CO moo (OmSOMmSO NWNO 


vo f>. On m CM ■**• mvO S0\0 h N * mvO N0\0 h o •t mvo SOO h « + 
cm cm cm rofOfOf<imrofOfO'<l" , t^"<J"*'t , t*ioio>oioioiom mvo vo vo vo 

ooooooooooooooooooooooooooooooo 



t^ m **• m cm 



OOO t^vo m CO CM m On t^vO m **• CO cm m o onoo t»»vo m •«*• CO 



O m cm co »f m mvo t^oo on o m cm co co ■**• mvo t^oo on o m m cm co <<*• mvo t>« 
t^t^r^f^t^r^r^t^t^t-* t-»oo oo oo co co oo co co co oo co o> on o o\ o\ o a on o> 
cocooocococooooooooocococococooooooooooooooooooooooooooooooooo 



ONt^in^CM ocovo i 
cm moo m • 



O txvO *$- CM 000 t^mcOW ONOO VO ■<*■ CM ON t^ lO 



, t^ ON CM mOO M COVO On CM IO00 COVO ON CM ■*»• f-» O COvO Oi H t S 
t^ t>s t^CO COCOOO ONONONO O O m M m cm cm cm cm ro fO M ■* * * n}- lO If) lO 
OOOOOOOOOO- 

VO NO NO VO VO "~ " ' 



5vovOvovOvOvOvOvOnOvOvOvOvOvOvOvOvOvO vO vO VO no no no no 



FUNCTIONS OF A ONE-DEGREE CURVE. 



303 



IMMHNNNCvlCVlCOCOrOrOrO-* 



VO O -*00 N vO O "*-00 N vO O -*00 NvO O ■*0»fONM in On ^00 « vO O m O 
Q N IONIAN *NO\H -4-vO On H rOvO CO H CO 1O00 6* rO lO t^. 6 N ^ N 6 N ' - 

on o> o* o> O O O O m h m m n n n w mfnfon't^^'*ioiflin »ovo vo vo 
r>* t>» t>» t^oo cocooooooocococooocooococooooocooooooooooococooococo 



moo h tj- t^ h 

w 0J 



■<*- t^. o ro t*. o -*t^H 



i moo N in o* N vo on rovo rovo 



Q 

o 

Q 



CO ts«vO in in •<*■ ro CM N m O On OiCO t^vO O iflitrOfOW m O O OnCO CO t»»vO vO 

M N CO Tf invO t^-CO o> O m M n co -4- invo t-%00 ON O w n co -f "*■ invo t^oo on 
inininininininin invO vOvOvOvOvOvOvOvOvOvO SNtNSNtstsNNNN 
OnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOn 



N NinmOOOvO "*■ m 0\N^N t^ u 



■i 00 vO "<*■ H ON tx •»*■ « 000 WM 



moo O covo o> m tj- r-^ o n inoo m ^j-vo on n in t^ o covo oo h rf ^ o n moo 

n n rommm^'ti-ininio invo vo vo vo t*» rx t^oo oocooo o>o>o>0 O O 

vO vO vO vO vO vO vO vO. vOvOvOvOvOvOvOvOvOvOvOvOvOvOvQvOvOvOvOvOvOvOvO 



ho N no «l_co ro co co ro ^ 



Q 



H -^00 N VO ONfONH -^00 NO On « N M \n On M N ^00 W vO ^"00 N vO 

M co inoo O N in t^. 6 N 4-t^OvM -4-vO Osm COVO00 M CO inOO 6 fOiONO N 
N N N N rOfOfOM't^^'t'tiniom tnvO VO vO VO ISNN t*»00 00 00 00 On On 

MrowotnrofOrommroroMroroMfOrofommncorOfOrorofommto 



v© 



m inoo h -^- 1> o covo o» n inoo h •>$■ t^ o COvO o» covo o» n in on n inoo w in 

n co -*vO t^OO M N CO invo t^ ON H co ■<*■ invO 00 o> O w ro TfvO t^OO H 
OOOOOOMWMMMMHMNNNNNNWCMrororororororO'>l-'<*" 

ro N h Onoo oo txvo to •*- co co N h Onoo oo t^v© in rh co co N w Onoo oo 

"*■ iovO t-» f^OO 0> M N ro •>*■ invO t^OO OO On O m W ro ■«$■ invO t^OO On On O «h 
cvi n n n n n n cororororororororororo-^Tj-"<t-Tf--<*-''*-'<*-TJ-Ti--<i-Ti-u->in 
OnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOn 



s ! 



vo ■«*•« MAfOH ONt>s.ioro000vo *« 000 lOrOH ONt^voronoOvo ■* 
M ^- t^ N IO00 w rOvO ONN lONo" cn^o ON m ">*■ t>* O N ^OO H ^vO On N lO 

^■■t'tminin iovo vo vo vo t^ t^ t^oo oooooo o»o>o^0 oOOmhhm«« 
NNNNNNNNNNNNNNNNNNNWNMrorocororororororo 
vO vO vO vO vO vO vO vO vO vO vO vO vO vO vO vO vO vO vO vO vO vO vO vO vO vO vO vO vO vO vO 



804 



FUNCTIONS OF A ONE-DEGREE CURVE. 






U 



h\o O to o\ •**■ <j\ cooo N ^ « <o h mo 10 -<f o\ «■*■ on cooo cooo cs t>. w tv- h 

00 O fOlOSO N lOt^O N lONO N »n N 6 N 4" t^» 6\ N 4" I>» 0> N 4- r^» O. N 

m ■* -t ■* •+ in m m invo vo vo vo t^ t^ ^ t-^oo oooooooo o> on o> o> O O Q o h 
onononononononononononononononononononononononononono o o o o 



On co r> m in o> co ^ h m on co t*. h lfl^MNH in o> co r^ m mansH m 

O •<*■ in fxOO C\H N + invO 00 Oh CI tO lOvO 00 OVO CM CO mvO t>. ON N CO m 

cm cm cm cm cm cm roronrorofOfO't-t't't't't'tioinminiriin mvo vovo^O 

CNCMCMCMCMCMNNCMCMCMNCMCMCMNCM<NCMCMCMNCMCMCMCMCMCMCMCMCM 



t*. two vo in in <<*■ rf- co co CM CM h i 



0> Onoo oo t*% t^vo vo m in *it- •«*■ co co co 



' 0> M CM CO ■«*■ IOVO tN>00 0\0 H N N rot invo C-.00 On M cm CO "*■ lOvO 
OhwhhmmhHHHCMCMCMCMIMCMCMCMCMCMCMCOcOCOCOCOCOcO 

ooaoooooooooooooooooooooooooo 



MOOvO rOHOO m CO 00 m« N«n« On rs. ■«$- CM OnvO •<*• h OnvO MHOOvO M 

h covo On cm •* t>s o co moo h -<i-<o Old •>*■ t>» O co moo h -<j-vo C\« iono co 
On On On on o O O h h h w cm cm cm cm rococo-^-^^Tj-xDinin invo ^o^o ss 
^**^-ioininmintoinioiniflioiniom>nio>nin>niniomininioin>n 

VO vO^O'OvOvOvOvO^OvO'OvOvOvOvOvO vO vO vO vO vO vO vO vO vO vO vO vO vO vO vO 



C* "«l-vO 00 0« "xhvO 00 CM ^vO 00 N "*vO 00 O « "*vO 00 N ""*-vO CO 

m m m m m cn cm cm cm cm wcocorotot't-i-^^inioinio invO 



CJ ^vO 00 N "<*-vO 000 N ""t-vO 00 CM -**-vO 00 O N -*»-vO 09 N <*NO 00 

M M H M H CM CM CM CM CM COCOCOCOCO'^-"4-'*'*'*miniOIO IOVO 



On CO t^ CM vO O lO On CO00 CM VO M m ON -<J-00 CO t> CM vO M in ^ ©v rr:oo tl Nh 

■<1- t^ On CM 4- t^. On h 4-vO On h 4-vO 00 m 
vo vO vo t^ t^ r-s f^OO OOOOOO On On ON On _ 
000000000000000000000000000000 00>OM>0\aa(M^O\OiO\0'0»0\a 

COMCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOfOfOCOfOCOCOfO 



00 



vO rt- (^ h inOO CM vO !>fOSO ">h00 m in On CM vO O fONN m o> CO tv h ioa 

H fO'tm t^OO 0>H N fO invO 00 ON 6 CM CO -^"VO t>» On M CO ^ »0 t^OO 6 H « 

00 00 00 00 00 00 00 0\0\0\(M>OiO\0 OOOOOhhhhmmm«N« 



WN<NWN«VJWNWC1NNWN«NW 



VO m't'tfON <N m OV30 00 t>»vO vO V)'<J'*fOf< N m O O00 00 r^ tx t* 



Ov m <N CO t*- invO t>.00 OO Os H W CO rt- invO t^OO Ov C 
t^OO 00 00 CO 00 OO 00 OO OO OO 00 OsO\0>OsOM3M>OsOsOst 
OvOvOsOnOsOnOnOnOnOiOnOnOnOnOnOsOnOnOvOsOnOnC 



i <n co ■*• invo t> 



CO H 00 vO CO H OwO "*h Ovt^-^-N OiMOM O t^iOcOOOO lOCOHOOvO COM 

00 M COvO 0\N "<*• t>s O CO IO00 M -4-vO 0> « lOOO O COvO OS H ■>*■ t^ O (M u">00 m 
O M M H M N N N COCOCOCO'^--<*-'<*--^-lOlO IDVO VO^OvO NN t^OO 00 00 00 0> 

vOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvO 



« "frvO 00 O 0) ■'t-vo WON '"hvO CO CM "<fvO 00 O N ^vOOO « "*vO <» 

h m m m h w w n w « c>cocococO'«*"*"^-'^'>*-»o»o»n»^ *n>0 



FUNCTIONS OF A ONE DEGREE CURVE. 



305 



h m m »o m mvo 



m ts w tsNoo cooo cooo •<*- on -<i- On ■*• m o no m t>» <n t>» cooo ■<*■ On •**■ O »o m 

t>» On N ■<*- S On N tJ- t>» On N ^so»« lONO w m N O N lO ts O n 1000 O CO 
oooo onOOOiO O O O hi m h m cm cm n rofOfom^'+ThTMninin mvo vo 

-*■ On mOO CM N«« H lOO ^ On ■<*- On -*00 COOO COOO CM t^N N« NN ts N t>» 

oo oiM « *io tvoo O h ntm t^oo O m co tj->o NO\0 n n mvd oo o> h cm 
O O ih h m m m h cm cm cm cm cm cm <n mrororommfOTj-Tj-'t't-tTj-'tinin 

mmOOOnOnOnOn Onoo oo oo oo Ms n fsvo <o no uiirimin't'+Tj-MfOcnro 

vOnO tsOO 00 On m CM ro Tf lONO t^OO O\0 h N M* *O\0 t>»00 On m CM CO ■<*■ 
nOnOOnOvOvO SNNNNNNNN tsOO CO 00 OO OO CO CO 00 OO CO OnOnOnOnOn 

ooooooooooooooooooooooooooooooo 

OnvO MO ts, ">*" H 00 NO COO N'tHOO ifl« OnvO no Nttl OnnO COO N+m 

■+N0 M u">00 M COnO On CM -4- t>» 0) lOCO CONO ON M M* tx On CM. VOOO COnO 
in u-jvO vO vO NO NtsN ts.00 COOOO>ONO\ONOOOOHiH<i-iHtCMCMCMCOCOCO 
vOvOvOnOvOnOnOnOnOvOnOvO^OnOnOnOnO SSNSNSSNtsStNNNN 
vOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOvOnOnOnOnOnOnOnO 



hhmi-imcicscs «-w fO(Otororo^'<j-t't-tir,»nio«n m< 



m vO hvO mnO hvO hvO h nO hvO h vO hvO hvO mvO m no h "O h vO mvO h 
CM ^-NONN -tNONN Tt-t>.ONW "ttsONN •+NOM -<1- ts, ON CM -t N ON N "tf- t>. 

m h m h cm cm cm cm commro-^Tt-Tj-Tj-mmu-) iomd <o no no snn t^oo oo oo 

ooooooooooooooooooooooooooooooo 



-«- On COOO CM NO H lO "*O0 CO t*s CM NO lO ON -*00 CO ts CM NO H NO lO «N -«- 

uto sono N n mvo oo ono n n mvo co on o cm co m\o oo onh « t iono oo 
no no no no nnsnnn t^oo cooooooooooo onononOnononono O 

NNNNNNNNNNNNNNNNNNNNNCMCMNCMCOCOCOCOCOCO 



fH 


COCOCMCMHHH.OOOnOn OnOO CO NN tsvO NO lOlOm^-^COCOCON N M H 


O 

Q 


NO t-»CO ON i-i CM CO >*> rj- mvO SCO On w CM CO tJ- mvO SCO On O h N CO tJ- \f) 

roronn'+'t'+'t't^<t^-Tj-'<j-'tinmmminio»ninin iono no no no no no 

ooooooooooooooooooooooooooooooo 



MO NWN OnnO COH00 ION On S -<j- H 00 lOCOO S •+ H OnvO CO MAN O 

cOvO 00 hi -*)-vO On N m SO CO lOOO H tJ-vO ON l^ S O CO XOCO m -<*-vO On N •* 
S S SCO 000000 ON <J\ On O O O O hi h hi hi N N N COCOCOcO-^-^-*--*t - vn»0 
inininminininioi/l ionO nOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnQnOnOnO 
nOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOvOnOnOnOnO 



306 




FUNCTIONS OF A ONE-DEGREE CURVE. 




J5 


« ^-VO 00 N "«*-NO 09OM ^-vO 00 C» «<f-vO 00 N rf-vo 00 N rJ-vO 00 O 
M H M h w CJ N N C* c* COCOfOCOfOTj-^^-Tj-^-lniAlOiO IOVO 





OvO H StOOMflM IN. N00 "*" O vO N0O ^»- vO MOO •<*- vO N 00 rhOvO WOO j 

O « msO « iooo O ro moo h rovo oo h Tfvo oh 4- i>» on w 4- iA. d N uit\ 
"t ■* 't * m ifl lO iovO VO VO vD t\NN It-nOO OOOOOOOO^iCTiOnOOOhhhh 


u 

W 


HVO 01 NNCO (O00 -tOMOHVO N N rO CJi •+ »0 H N W 00 tOOMOO^O H ts. ' 

oo on m n ■<*■ ir» t^od o w ro iovo oo o\h « n-vo sao c! m iovd oo d h ro 4* 
ONONOOOOOOHMMMMHMWNWNWWrommrofomTj-Ti-^-^ 


p 

o 

p" 


o>oo oooooooooooooooooooooocooooooooooooooooooooocooooooooooooo 


d 

M 

O 
33 

u 


vO ro Ovo fOOvO roo NMO NrtO i>» ^- i>* ■«• h N'tnoo -tnoo ■tHOO 


vo Oih tso « iooo o rovo oo h -^-vo on ■<*• t^ o w iooo o rovo 00 H n-vo 
h h n n n roromro-^-Ti--^-Tt-ir)iou-) iovq vo^o nnn r^oo oo oo oo a>o>o> 
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 
vOvOvOvOvOvOvOvOvOvovOvO vO vO vOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvQ 








z 

s 


m m hi m h n n w « n coromcocO't'+'*'*-+iomioio iovo 


Q 


h vo N i>» rooo •**• in h vo <m tx rooo tj- o io h in. rooo •<*- io h t^ rooo ■>*■ 


ro moo ro iooo m rovo oo m rovo 00 H »d-vo 0> H t}-vO 0> <N *NOi« "*t^O 
vOvOvD t^ t^ t-<. t>»00 OOOOOOOONONONOOOOHMHHNWNNrororo^- 

MMMHHHMMMHHHHHHN(NO)N(NN<NNWNWNNrNNN 


u 

W 
1/3 

X* 

W 




Mrorororororororororororororororororornrororororororororornro 




P 
« 

o 

p 


M« N N IN CM H M M H H H OnO > >OnO>OnO > >0 > >0>ONOvO > (OvO v >O x >ON 
■«- iovO t^OO 0> m N ro ■* IOVO t->00 0> O m CJ ro •»*• iovO t^OO Ov h N ro 

0>ONO>0>0>C?»OOOOOOOOOOOWHMMMHMHIHH(NNNN 


d 

OS 

o 

X 

u 


H 00 lO H 00 WN OvvO N OvvO COO tvrOO ts^HOO *J-h00 ION OMO IN OnvO 


vO oo m rt-vo Ov N •**- i>. w iooo h rovo <y> h t$- i>. ct> n mtvO ro »ooo h rovo 
mrO't't't'+iniO iovO vo vO vO t>. t^ t^ t^OO OOOOOOOnOONOOOOhhh 

i>.t>it>.t^i>»t^t^f^t^t^t^i>»t^t^i^t^t^t>»t>.i>.t^i^t^ t^oo oo oo oo oo oo oo 

vO vO vO vO vO vO vO vO vOvOvOvOvOvOvOvOvQ vO vOvOvOvOvOvOvOvOvOvOvOvCvO 


55 

2 


« •*^0 00 O N ^VO 00 O N -"ihvO 00 O CI "^-VO 00 O W "^-vO 00 O N ^vO 00 Q 
H M M M H « W W C4 C4 COfOrOCOCO->*-^-'*'*'<-lf>lOtOlO IOVO 



FUNCTIONS OF A ONE-DEGREE CURVE. 



307 



I H«« N N N tOCOtOncO'" 



S^h N'+O NMO NroOvO ro0"0 roOvO roO smo N ■**- o N -"il- m n 

VO 0\M ■*■ N 6 C\> "">00 O rONO OO w -<1-vO ONN "*■ N O N IHOO O rOvO 00 w -«->0 
On On O i-i h h m <n cm N cs> mrororo-<*--<*--^-ir)ir)ir5 invo vO no no N N N 



i NfOOvO MOO -*0 NfOOvO ro On m N CO 1/1 h N "*- t>. CO OnnO N On in 



N -<*- in N On O m co in NCO N fO u-)\0 00 H CO mvO 00 O hi m -^-vo 00 On M 
o>ONOOOOOOOOOMHi-iiHMMtNNWNN(NmfOrofomrorO'^- 



n w w n n ffimfOMrt^-^^'j-'Tinminin mvo no vo no no tx 



«*■ mvo noo on O m n m ■<*■ mvo noo On m n m ■* invo noo On m cs m Tt- 
oocooooooooo onononononononononono o o o o o o o o o m m m m m 



^- t^ fO OnnO N OMflHOO "* txro ONNO w omdhoo ^j-o NMOvO cni om 
vo o h tj-vo ow "*■ n o n moo o co moo h cono on w Tf now in no in id 

N NCO O00O0O ON ON ON O O M M H M N N CJ CS Mf f )fOfO^-'t'tl^ifltn 
OnOnONOnOnOnononOnOOOOOOOOOOOOOOOOOOOOOO 
vOvOvOvOvOvOvOvOvO NNNNNNNNNNNNNNNNNNNNNN 



oomM sro on>o moo -^-o sroOMflHco -*m srooo mOMOSOO i«H n 

NO fO iriOO O rONO 00 H Ti-vO 0\H tj-nonn in N O ro mOO O f^vO 00 M tj-vO 

m cj n w n nnnn't<t*'tiflmin \n\o no no n n n noo oo oo oo on on on 



u 

W 


SNOO tJ-0vO H SMOMflOvO CN 00 tJ- ON m w »n. ■**■ NO NOO -<i- SrOOMfl 


t*-mO M^H N rh in NOO CJ CO lONO 00 On h CO tJ-vO OO ON m N -^-vo N ON N 

****ioiflininm "">no \o no no no no no nnnnn noo oo oo oo oo oo on o< 



0O00O0O00OO00O000O0O ON ON ON ON on o> ON ON on o> o o o o o o O O h 



oo *hoo •<*■ h N •<*■ sroOO fO OnnO N OnnO N on in N 00 in h 00 •* m N ■«■ 



mhmmmwnmnci mntomto^ 



FUNCTIONS OF A ONE-DEGREE CURVE. 



MMHHMNNNNNCOCOCOCOCO'"*-'' 



oo toroo Mfl« ONt^-^-H onvo roooo »o co oo 10 co o oo mmooo iriroo 
t>s O covo oo h -<fvo <jn mso covo oo h •* tv o n moo o covo on h -4- t>» d 

u-\\0 vO vO vO t^ t^ t^ f^OO OOOOOnOnOnOnOOOOhmmNNNNcOCOCO-"*- 

minioioininiominifliominioin iovo vovovovovovovovovovo<ovn\ovo 



N'+HOO ION OnvO MO N't H00 IT) fO N ■<*■ h OnvO MO MON ONVO CO H 

H CO lOv© OO O H fOlO N00 N fOlONdO N ^wiNOh N -^-v£> (>. o> m co 
OnOnOnOnonOOOOOOhihhhhhnnnnnncocococococoti--^- 
in uiuim iovo vqvOvOvOvOvOvonOvOvOOvOvOvOvOvOvOvOvOvOvOvOvOvOvo 

^OVO N t-*00 CO00O\O\000HHNNNfOrOt + tm iovo VO N t^OO 00 ON 

iovo N00 On h N co iovo t^OO On h N co -*■ iovo N00 On h N co t»- iovo 
t'+t't'tioinioininmmin iovo vovovovovovovovovo t^t^t>»t^t^t>st^ 

NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN 

vO N00 «4" vO N00 ^"OvO N0O TfOvO MOO ■* NO WOO ■<*• VO N 00 ^- 

"tvO ONH ^*- t->. ON N "rf N N lO N CO lOOO O COvO OO H COvO On h t}-vO On N 
rOfOfOtt'ttiniO iovO vo \Q vO N t^ t^ f^OO OOOOCO OnOnOnOnO O O h 

MHHHHHHHHMHHHHHHHHMHMHHHHMCNIINCNICMN 
SNMsNNNNNNNNNNtNMSNNNNtNNNNNNNNNN 



n-*hoo lflN o>vo mo smo NtHoo inno stHOO mo o n^hoo 

VO ON ■*■ N CM ir>o0 H COvO 0\H tSO^N lOOO COVO 00 H t NOiN ins 
N t^.00 0000 OnOnONOnO O O m H H h .N N N fOfOMfOtt't'tiniOifl 



ION On »0 N On lO N ONION ONtniN ONION ONlONOO tHOO ION OnvO CO o t- 
H CO "<*-VO 00 ON H CO "tf-VO 00 ON M CO tJ-VO 00 ON HI CO Tj-VO 00 On H CO rt-vO OO H 

t "t -t t t tin io io in ui iovo vovovovovo t^t^t^r^t^ noo oo oo oo oo on on 

I0l0l0l010l0l0t0t0l0l0l0l0l0l0l0l0i0l0l0l0l0»0l0t0l0t0l0l0l0l0 



Q 


tx t>» t^CO 00 00 


"OnOnOnO H H H N N N COCOCO"^-^-»!j-l010 IOVO vO VO VO 


O 
Q 


>f IOVO f>.00 ON O H N rfr IOVO t^OO On H N CO tJ- lOvO NCO On H N CO rt- lO 

H H H M H Hi N N N N N N N N N MrOfOrOMMrOfOrOfOt't't'ttt 
NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN 




Q 


mn NfOOvO 


NOO -«J-0 t^COONVOH f» »j- vO 


NOO "<*- H SfOOUIHOO 


•«*• o 


(A 
O 
8 
U 


lOOO O COVO 00 H COVO On h Tt-vO OnN -*• N O N ION0 CO lOOO COVO 00 m -f 
lO tOVO VO VO VO N t>. r-* NOO OOOOOOOnOnOnOOOOhihhihiNNNNCOCO 
00000000000000000hhmhihihihh.hhhh.hihi 



N ii-vo OO ' 



J 



FUNCTIONS OF A ONE-DEGREE CURVE, 



309 



u 


MflMH OwO ■*■ N 00 vO ■*■ N 00 vo "*■ N On t^ to co h On N. u-> co m 


m 


\0 x>. o w co ^t-vo oo N romso\H w -*vO CO O w nmNO»0 N -«*-vo co 
0\OnOnOOOOOwhmmmmc*nnnncocococococO'^-ti--»j-.<*-ti-u-> 
vo vo vo t>»t^t^t^r^f»txr^tN.t^t^r^r^t>»t^t^t , »t^t>.t^t^t^^fxt^tN.t^t'^ 



«* *vO 00 « *"0 CO N "<*-vO CO N -+v© CO N "*vO CO N ^-vO CO Q 



•^•W 0CO NtOCOH O^MflfOH ocovo •«*• n H On tvvO ^-N 0>Sm'+N 

COvO 0\h •«*■ t^ co u">00 m ■* t^ O N tOOO h -<j-vO On N tOOO m COvO On N iO00 
N N W COfOfO-^-""*-'^--**^"^ lOvO vOvCO t>. t-^ t~* t^OO COCO l>(M>00 O 



XT) lOvO 


t>> t>.CO On On w 


M N 


co co ■* in tnvo 


t^ ^00 CO On 


H N N 


CO -^ lOvO 


CO On 

O O M 

MMro 


H 


N ro^-in t-»oo o\0 h n mt tnvo t>»oo On O h co "*■ invo t^oo & 

MWHMHHMNNNNN««NC'JNCOCOCOCOCOCOrOCOCO*<1- 

MfOfOforomfOMroromfonrofOfOfororomforomfOfOfoco 



iflHvO N t^ CO On "*■ VO H t«s CO00 ^ONWNHVO N 00 CO On ">*■ vO H t>. CO00 ■**• 

0\N *SO\N "*■ t^ O N WNS0 N UNNO CO IO00 CO »O00 M COVO CO w COVO 
co OnOnonOnO O O h m h m n n n n mroroM'*'<j-'<j"*io»nuN invo vo vo 
N N n N N cococococococorococococococorocococococococococococo 



0COVO CO h On t>» ■<*■ N t>» in f» CO VO "<*• N On tx»n CO M OMAN OCOvO ^ 

O <N lOCO H covo ON N int^d COVO CO M 4- tA. ON N tOCO M COvO On cvj inNO CO 
■+ ■* ■* ■* m lO lO lOvO O^O NSN f^CO OOOOCOOnOnOnOOOOhmhWCS 
vOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvDvOvOvOvO t^f"»t^.^t^f^t^t^t*» 



HOOvO cohcOvO cohcovO cohcovO cohcOvO con o*n<omhoo\o ■<*- n o 



i N ^VO 00 ON H I 



I m f^OO N "*VO 



vOvOvOvOvOvOvOvOvOvOvOvOvOvOvovOvOvOvOvbvOvOvovOvOvOvOvOvovOVO 



»00HHSNM(O + *ifl tOvO vO t^ tv00 COONONOOHMNNCOCOrJ-v) 

M N CO 4- lOvd t^oo 
■ wOOOOOOOO 

NNNNNNNNNNNWNNNNNNNNNNCOCOCOCOCOCOrOCOfn 



0vO N t^cOONtOM f> N CO ■<*• vO N N M OMft H fx N 00 "<*■ vO N tx CO On (*j 
N "■*■ t>. ON N ■«*• t»* N VNt^.0 CO lOCO O CO 1O00 H COvO 00 l 



i « NN M M rtfOtOfOCOT 



310 



FUNCTIONS OF A ONE-DEGREE CURVE. 









00 




O « *^O0 O N ^t-vOOO O W "«*-vOOO N ^VOOO « ^vOOO O N -^vOOO o 

m h m h m w e» n « w rocofOcoco'«*"«^'^"^-«4-«oin»o»r> mvo 


< 


OOO t»»vO M- CO N M OOO bsvO W <«J- N « H OOO t>.vO VO ifli-Mtl M O 

covo O N moo h ^- t^ o* N inoo h rj- r*« o co\o oo h ^j-no covo o n inoo m 
O O O O O O m w h h W 0> N cococO"*-^-T}--<*-ir>in mvO vO^OO NN t^OO 

oo oo oo cyiO^o^o^o^o^o^o^o^o^oo^o^osOso^oo^o^ocyiOso^o^o^oso^o^ 


u 

w 
m 


in co « m o ooo vo w^« h o ooo t-^vo «*«« h o ooo t^vo in ■«*- co co 

insoiH m -4-0 MOM -<*-vo oo o>m ro in s a h co in l>.00 N "4-VD OO N 

OOOMHMMMWMnNWNCOCOfOCOCOrJ-^^^-Tl-ininiOio idvO VO 

oooooooooooooooooooooooooocooooooooooooooooooooooootooooooooco 


Q 

K 

o 

d 
2 


COOOmmNCO'*-'* lOvO t^ t^00 O h N CO ■»*■ invO t-xOO O M « CO M- W 

w co in<o t^.00 o\0 h n n* invo t^ o o m n co -»*■ invo t^oo O h w co "*■ in 
t^ t>. t^ r^ i>> i>» t>*oo oo oo co oo co oo oo oo oonooooo^ooo o o o 

COCOCOCOCOCOrOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCO-^'i*"-"^-"^-'^-'!*- 


d 
« 
o 
a 


b<» M t>» CO00 COCO ■*• O ^* O **■ O WO mnvO HVO N t^W b<» COCO COOO CO o ■"*■ 
w lnisO w wso n mso CO IO00 O CO lOOO CO IO00 O CO »O00 O CO IO00 

t|- •* tmo >n in invo vovovo t>» t^ t^ t^oo ooooooooooooooihihhm 
^^^Th^^^'^j--«<j--»*--^-^-M-Tt-T*--^-Tt-Tj-Tt-'«t-T*-^Tj-ir>io»ntoioioioin 
t>.^i>»i>.i>.t*^^i>*t^i>»i^«^^i>.i>.i>»i>»i^r>»i>.i>.^^t^i>*t>i^t>.i^t^ 




W M"VO 00 M ^vO 00 N -"t-vO 00 N -*J-vO 00 N -*J-vO 00 N -xJ-vO 00 

m m m h m c* c« « n n mrocomM't'ft't'^intninin invo 


o 

00 


fc 

£ 


« fvOCO « -*-vooo N -^-vooo n -«*-voco N *J-vO00 N ^-vOOO 
m h m m h e» n n n n MMfonM + ^'t't'tinmioin invo 




O t»»vo -troH ooo t^m^-N m onoo swiM-mn o ooo vo m ^ co h o o 
co d covd o\ w iooo o covd o\« mso co\o o> w moo d covd d> ci »ooo h co 


a 


O H COIOK.OH N "4-vO 00 N fOlOSO\H CO io«o 00 N -^-vo oo o\ H CO in 

in m m in in mvo vovovovo t^r>.t^t^t^ t^oo oocooooo ooooonoo o O 




d 
O 

Q 


O t>« r»«oo oO w N flfOt invO iO t^oo OOnOh n w cochin invO t^co oo 

O m w co 4-vO MMOO m w co-* in>o r>»oo oh n ro + invo i-^co <j\ w n 
'♦'♦^■^•'♦•t't^tioioioininmminin iovo vo^ovovovovovovo t>.t^t^ 
corocococococococorocococococococococococorocococococococococo 


* 

o 

3 


♦ OinOvO m ^N t>. cooo ^CMnO mnvO N l^ cooo coo-^-o mnvo m t^» 
yQ oo h 4-vd o\ w 4-vd d\ h -4-vd c>n -4-i>»c>n 4-i>»c>cj -^-1^0 w mi>»o « 

OVO NSN 1^.00 OOOOOOONOOOnOOOOhmmhNWNCOCOCOCO'"*--^- 

cOcOcococococorororococororO'^^•^^-^^'<*•'<*■■<^"<t•^'<^x^'<^■■<*•'^- , *•■*'^■■^■ 


fc 

8 


e» -*vo 00 o -^vo won -*no 00 <m ^-vo won •^•'O won ■^■^o 00 
m m m m h n « « « c< cococococo-«*"r'^''>f'^'inininin m\o 


I 







FUNCTIONS OF A ONE-DEGREE CURVE. 



311 





1 

•• 

w 

90 




« ^t-vOOO « *-vOOO N *vO00 N <*vO 00 O N ^-vO 00 O N "*v© 00 Q 

m h m h h n w n « n cococococo*"*"*"*'"*'wwww wvo 


Q 


♦ "♦♦^^^fOMfnfOMfON W N N N N N N N IN N « N MMmtOMfO 

d> « woo h ^so covo o\ w woo Mtiso covd a. n woo m -4- 1^» d covd on 

« NN t^OO 0000 Ov Ov Ov On O O H M m N W PJ P) M tO rt ^"t <t lO m lO lO 


u 
u 


xovo^o u^i^in^-^-"*"*'^^^^^^^'^^-^'*^^^^^-<*--^^^io 


O N -^-vO 00 O N **-0 00 O N -*-VO 00 O N "^vO 00 O N -<fvO 00 O N <*-vO 00 

n pi w w pj MforofOfO'+'f*^-'<Mninioin wvo <o vO vo vo t>» t-*. t^ c-» t>»oo 

OiO*O^OiOiOnO^OnO*0<O^O^O^ONOnO>O^OiO^O^OnO^OiO>0*0^0*O^OiO>0^ 


Q 

« 
O 
Q 




^^^^^^^•^^^•^■•^•^■^■■^■^■^^■^-■^■^■^•^^■^■•^-Tj-Tl-M-^^ 




Q 

OS 

o 

X 
U 


vO HVO M v© WO iOO WO iflO WO > i'**"ON'*0'*Ov , 'i*-Ov -*00 rOOO CO00 CO 

COVO 00 H COvO 00 H COVO 00 w COvO 00 o co woo O CO WOO CO woo CO woo 

OnO»o>0 O O O m m m h m pj n n fOfomro^^^^minm wvo vovO^O 
ID io wvo vOvOvOvOvOvOvOvOvOvO vO vO vOvOvOvOvOvOvOvOvOvOvOvOvOv©vO 


55 


m m m h h n w « w « cococococo , **"*"'* , '«-wwww wvo 


00 

£ — 


55 

i 


N *-v© 00 N *vO 00 N <*vO 00 N *vO CO N *vO CO • N ^vO 00 

h m h m m n « in « « mmmrtrtt^t + ^wiflinifl wv© 




ovoo oo t>. two min*tflfONNHOOOO» 000 oooo n t^vo vo\omw* 

m covd a» N woo m -4- t^ d covd on n woo h covd d> n woo h tNO covd o> 
cooocooo o> 0> 0> Q Q O m m h h p> « N ro n ro m ■♦ * ■* in io iov5 vb so ^O 

(j(MMJ&(M^0OO00O0OOO0OO0OOOOO00O00 


8 


CO N K H 0^ O^00 t«» t>»vO WW-^-POrON M m 0^O^ On00 00 00 NN t>vO 

vOvOvOvO SSSNS t^00 MOOOOOO (JOM^OO^OOOOOhmhmhN 
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOCO 00-00 COCO 0\0>CT>ONONOOiO>C7NCyiCyi 


Q 

OS 

O 

Q 


wvo t^oo o>0hn m + wvo txoo aOMN«* wvo t^oo o o h n po ■*■ w 

WO t^CO Osm N m* WVO txOO On O W CO * WVO t>»00 0> H PO •* WO t»»00 
OOOOOMHMMMMMMMWWNNWWWCSNrorororororororo 


Q 
05 
O 

3 


<r»*O>+»*OMn0 WO W0 WO W0 WO W0 WmvO mvO mO hvO 

00 O n woo ro WOO h covo oo m ro\o OO m mvO OO w COVO 00 m fO«o 00 h CO 
h nn n n mronto't •^■M- * w u-> m mvo vo vo vo t^. r^ **» t*»oo oo oo oo o o> 
mwwwwwwwwwwwwwwinwwwwwwwwwwwwwww 


55 


« "♦VO00 « •♦vOCO N -*vO00 N ""J-vOOO O N -*vO 00 N -*vO 00 Q 

m h m h m w « « « e» mtnnfon + *'rt + ininin<o wvo 



312 



FUNCTIONS OF A ONE-DEGREE CURVE. 



55 



IS 






I H M« N « M « MWfOMCO^ 



VO VO tsOO OiOOH«tO*t IOVO t>. tvOO 0*M N tO + vovO *n00 On M N CO 

O COvO On N vO ON N vooo H -4- tx 6 COvO 6\«\d <J\M vooo m -4- tx O 4- tx d CO 
VO VO VO IOVO VOVO NN C^OO 00000>0>0>aOOOHHHNNNrOrOfO** 

loioiomvoiomioto vo m loioioioioioioioioioioiooioiOLomioioio 



00 WO O h « « to^t io<o t>.00 On m N co ■<*■ vo t^oo On h N ro -«-vo txoo 

M fOvd 00 O N 4-VO 00 6 N -4-NO 00 H r?l Ifl N ON H fOiOt^O <N 4-vO OO 6 N 4 

■+■+■<)• >t io ui io m lovo vo vo vo vo nsnn t-^oo oooooo gNO\o>g\ONO O O 
OOOOOOOOOOOOOOOOOOOOOOOOOOOOmmm 
N«WN«WMWNMCN1«NWWN«NNWNMWWN«M«««« 



•**■ vovO 00 On N CO -*vO t^OO H W -<f iovO 00 On N CO -^-vO t^OO h Ct CO 

VOVO t^OO On h « CO Th IOVO t% ON m N CO -** lOvO 00 On m W CO -^-vO t^OO On 
OOOOOMHMHHHHHNMNWWWNNNCOrOCOCOCOrOCOCOCO 
lOlOlOlOlOlOlOlOtOVOlOXOlOlOlOVOlOlOlOlOlOlOlOVOlOlOlOlOlOlOlO 



■*oo cor^Nvo H IO0 ^-On COOO N NHViO+W COOO nn^ 

N 'J-SOiOl •<}■ v^ On N -^-VO On m -rfvO On m rJ-vO 00 H COvO 00 h 
"«*• ■<*■ M" ■**■ lO lO VO IOVO VO VO VO Ntsts t^OO OOOOOOOnOnOnOnO^ 

txtxrxtxtxtxtxtxtxtxtxtxtxtxtxtxrxtxtxtxtxtxtx tvoo oo oo oo oo oo oo 
txtxr-xot^t<Nt>.txcxtxt>.rxtxtxtxtNfxrxt*»^txt>»t>itxtxt'Nt>.txc>.t'»tx 



VO »n On ""*-00 

IVO 00 O CO lO 

O O h - " 



I M OJ N N N M cococococo^^"^^-^-ioioiom iOv£ 









2 

3 



I H 0) N « « « fOMCOCOn^^' 



■>COCO'^"^"^'iOiO IOVO VOVOVO NN t^OO OOOnOnOhmNNCOCO^vo lOvO 

Q o>« vooo* h 4s6 covo on eJ vooo m -4- t*» covd 6 covo on n vooo h 4no 

IOVO ^OvO NN l^OO 000000 OnOnOnO O O m m m N N <M (N CO CO CO •*»••**• ^ VO 
MMMMMMMMMMMMMMCJWNNNNNWNPJNWWNNCJOt 

<5 V0U^V0»OiOi0»Oi0i0V0V0l0l0»0l0iOi0V0i0iOiOiOiO»OiOt0iOiOi0VOVO 



lO vo vovO vO vO t«s f*. txOO OOOOOnOnOnOOOOhNNCO-^"^- lOvO vO t^OO 00 

O W <*vO OO O N ^VO 00 N -^-VO 00 H C? VT) tx On M CO vo tx ON M MIONOH 
OOOOCOOOOOOnONOnOnOnQQQQQmmmmmNNNNNCOCOCOCOCO'*- 
OnOnOnOnOnOnONOnOnOnOOOOOOOOOOOOOOOOOOOOO 

HHHHHHHHHHdNMNOONNNMNCINNNMNNHNN 

00O\Oh«+ vovO txOO O m o» co -«*-vo t^oo OO « ro + iovO 0OUOh« + 

h « •"*■ IOVO' l>»00 ON O H CO -4- vovO t>»00 O* O w CO ■* vovO t-^00 On Q N CO **• vo 
NNNNSNN t^OO 00000O0OO00OO0O0 OnOnOnOnOnOnONOnOnO O 

■^■'^•'^■'^•■^••^••^■'^-■^■^'^Tl-Th-^^-^- ^Tj-TJ-^-^-^Tj-Tj'^-lJ-VOVOVOVOVO 

COOO N NNVO HVO M lOO voOntJ-ON COOO COOO « NN\0 mvO lO vo On •«■ 

00 c5 CO VOOO C« VOOO 6 CO VO tx Q N VOtxO N IT) ts O N vOt^O N UN N QN N 
^O NNS t^OO OOOOOOOnOnOnOnOOOOmmhhWNWNCOCOCOCOCO'<<- 
vovovovovovovovovovovovovo NSSNNNSNNMnNNNNNNN 
rxtxt>.t'Ht , *tNtxcxtxfNt>.t<*txtxtNt%t'»tNtxtxtxt'>.t'Ni«HtxtxrxCNtxt>tx 



CJ ^"VO 00 N ^vO 00 N -^-vO 00 O N ^vO 00 N ^-vO 00 N -*vO 00 Q 

H MM M M W W N N N COmfOmfONj-^^^^LOUIIOUN VON© 



FUNCTIONS OF A OXE- DEGREE CURVE. 



313 



« ^vOOO « *vO00 



SM -*vO00 CM "*vO00 N ^"VO 00 N *vO 00 O 
cm cm cm cm tomncorof't't'^^ioinioin iono 



mNO^H m ^vo oo n ■* in ^ onh co in t>» On m co in t^ On h co in t^ On m co 

. »4-00 m ><i- t>» ■*• I s * cono 6 co 
IDVO \OvO NN t^00 OOOOOnOnOnOOOhwi-iCMCMCMcOCO 

mininininininininmminininininininininininininininininininin 



wjnO oo cm ^- in r^ on w co in t>. On n co moo cm -*vo oo romtso^H covo 

, . . MlONO CM "<*-vD On m CO in t^ CM "<*-nO 00 M tOifl 
ONONONO>OOOOOi-iMiHMNCMCMCMCMrOCOCO 
ihmwhihmmi-ii-ii-ihiwm»-iCMCMCMCMNCMCMCMCMCMCMCM<NNCMCMCM 
NCMCMNCM«CMCMNCMCMN«CMCNCMCmcmCMCMCSCMCMCMCNNCMCMCMCMCM 



NO 00 ON H CM 



h m ts. on h CO ^-vo t^ On « CO invO CO Om « 



ro -<t- in t^oo on m n ■*• invO t^oo o\0 « m* mvo t^oo O w cm co -"4- in t-^oo 
l>» tN. f>* t^ t*s C^OO OOOOOOOOOOOOOOOO OOOnOOOOOO o o o o o o o 

~-" > <o NO NO NO NO vO NO 



inininininininint 



1 in io in io in iono no no no no no no m 



in on co ^ h m o "*oo cm no o -*-oo cm no i 



ifiOroNH in On co tx cm no *00 



oo o co moo o mioNO cm in t^ on cm 'J-nonh "t\o o> h cono oo m cono oo 
ooONOiONONOOOOHiMMMMWNNWcocorocOTi-^Tj-^mmm mNO 
0000000000 OnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOiOnOnOn 



i m h m m cm cm cm cm w tofococoro^ + f^'tiniriiom mv£ 



« ^-nO 00 CM ^nO 00 ( 



cm cm cm cm rofOfomro*****iniflinio mv£ 



CO ■*• m t^oo O CM CO ■••■no txoo Om N *m t^OO m CO -^-nO CO Oh N*m 
CONO On N mOO N mOO M -^- tx Q CO t>» COnO On N NO On N m00 M -*00 M -»J- t>» 

•«*■ ^- -^ m m mNO ovo ns t^oo oooOOnOnOnOnOOOmmmcmncmcococo 

COCOCOCOCOCOCOcOCOCOCOCOCOCOCOCOCOCOCO'^-'«*->^-«t-Tj--^^'<1-'<l-Tj--«J-Tl- 

mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm 



W 

in 


CO On h N 


■>*• mNO oo on m n co mNO oo on o cm co m cvoo o n tj-vo noh min 


argsrs 


HUl SOh ThNO 00 CM -0-nO On m CO m t^ On CN1 rhvO 00 CM in t^ On 

m w ii m cm n cm cni cococococo-«i-Ti--4--^-rfmmm mNO no no no no 

N«NNMNNCINNNMNclNNMNNMNN«NnNci 



CO ^-NO ^ On M CO ^-NO txOO M CO rj- m t^OO M CM •**• m t^OO On M N -*vO 

on o m n co mvo rxoo o» O h co Tt- mNO t^oo o»h n rot iono r^oo o m cm co 
co-^-Tj--^-T*-Tt'-^-^-^-ri-in mmmmminm iono no no no no no no no t>. t^ t^ t>» 
mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm 



CO W NO m m On CO00 MNO •^■ONCOt^HNO ^-00 fONMiflO ^CO cm no m in 

mOO O tOiflSO CM mt^O CM tJ- f^ On CM f SO\h >«J-nO On h tJ-vO 00 m CONO 00 
h h n cm cm cm mrOfOrott , tt"*ioinm iono no no no ^ t^ t^ t>.co co oo oo 
oooooocooooooooooooooooooooooooooooooooooooooooooooooooooooooo 

tsCstsNNtsM>tsNtN(sNt\t\tNtNt>.NtsC\tNt>NNtst>tNtNCNt> 



« ^NO 00 CM -^-NO 00 CM ■<*■> 



CM ^NOOO CM -«*-NO 00 CM ThvO 00 CM *vO00 N ^-nO 00 

i-i m h m cm cm cm cm cm cococococot'*-*'«*' , *mmmm mvo 



814 



FUNCTIONS OF A ONE-DEGREE CURVE. 



55 

3 



Q 



00 H *N0 COvO OiH *N0 COvO ON N 1TJ00 « »O00 M n*-00 H ^ t«. <* t>. 
■<*■ t»» O ^"ts.0 rot^O rot^O covO covd O covO (" 
vOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvC. _ 



vo on N moo h ^t^O co t> co in co i*» co r»» tONO co ^ o co ts. o ■* 



00 



i co m t> on o « ^t-vo oo on h co m t>.oo o « -**-vo t^oo o oj ■«*- m t> on m co 



■> ■*■ mvo t^.oo o m n co •«- u-> t»»oo o\0 m «* invo r>.oo o h « co "<*■ in t^oo 
.■'f't^'+^ioinioioinmioio mvo vo vo vo vo vo vo vo t>.r^t^t^t^r^t^r^ 

vOvOvOvOOvOVOvOvO vO vO vOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvO 



■*-oo h in on oi vo o -^-t^H moo cj vo on co t> h -*-oo « moi mvo o "froo h in 



o 

X 
V 



« ^ t^ o> h -<*-vo ON h covO 00 

oooooooooooo 
oooooooooocooooooooooooo 



romNO co in t>. On « -«l-vo <y> m rj-vo oo h 

IVOVOVOVO t^t>.t>«t^ t>*00 000000 ON On 0> On C 

ooooooooooooooooo* 
ooooooooooooooooooooooooooooooooooa 



55 

3 








<5 



rovo 00 m co inoo N m t> on N ^t-vo On m rj- f* on n ■*t^0 N moo co inee 

COVO On COvO ON N vO On OJ moo « inoo h inoo m -4-00 m -«-0O h ■<*• t>. m >«f t> 
cococo^'<t , ij-tr)»n mvo o^o nn tvoo ooooonononOOOhwhwoiwco 

»nininmininininuNininininvnuN»ninmioio »nvo vo vo vO vo vo vo vo vo vO 



vO 00 H COvO 00 H COVO 00 H ^-I^OnN in tx On N inoo COvO 0\« 'J-tsO CO.O 

mso s "d-vo on h co inoo o « tj- t^ on m covd oo o co in r>. on oj -^-vo on m cp 

COCO'<!h'»i-T»--^--«iJ-lA»n»0 invo VO vo vo vo t>. t^ r^. t^OO OOOOOOOO ONONONONO O 
WNNNNNNNNNNWNCNNNNNNNNCJNCNNNINNNCOCO 



X 



oo 

00 



« tm t^oo o n co mvo oo O m co -^-vo oo on h n -xi-vo t^ on o m •<*- in t>»oo o 

00 (JO H N t lOVO t^OO 0\ M N CO Tf invO NO^O H N to tJ-vO t^OO On M CO 
OOHHHHMMHHMNHNNNNNNroMfomnmnntottt 
vOvOvOvOvOvOvOvOvOvOvOVOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvO-^ 

00 NVO O COtxH in On CO tx h -rt-00 N vO "^00 h in On CO t^. M in on n vo Q •*• 

O CO moo 6 N lflNONN -4-t^C3NH -4-vd On h COvO 00 O CO lOOO O « in ^. O W 
VOVOVOVO t>.t^t>.t^ t>.00 0000000\0\0\OQggOHHHM«NNNrpfO 
OnOnOnOnOnOnOnOnOnOnOnOnOnOnOnONOnO O O O OOOOOOOOOO 
tNt^t^t^t^txt%t>txt%t>.t^t^t^t^t^ t^OO 00000000000000000000000000 



o 

X 



2 



« ^"VO 00 N "«fvO OOON -«-vO 00 N 'J-vO 00 N ^-vO 00 O N ^vO 00 Q 
m m m m m w w « « N cococococo^f^-^^-inininin »ovO 



FUNCTIONS OF A ONE-DEGREE CURVE. 



315 



O M ^VO 00 N "4-"0 00 O M ■ 



Q 



OifOSH lOO\fONH ir> <J\ rooo N^O ■* On r<-) t^ N VO -i" 0> fCOO N Nh^O 
-4- t>. w ^-tsH rj-00 11 ^-00 m mOO C\> IO00 M IOON*0 ONVO O r^.vO fO 

mfnm'+'t'+io'O iovo «« nn t-^oo oooooooNOOO>-i'- | i-iNc<rom 
oooooooooooooocococooocooooocooooooooooooo OOOOOOOOOnO 



M ID "<*-00 N l^ •■ 



} O ^O-rONN^ i 



in moi + o* rooo mnn t^ n t^ 



mNO N -t NON -tf- f>. On w tj-\o On m ^-vO On h rovo 00 w rnvo 00 m rovo 00 

ti-inininio invo ^o^ovo i>nn t^oo www oo^o^oo o O O h h m h 

NNMCMNC»NNW<NWCS1NNNWN<NC^NNNC<WC'ICSC)WN«N 



00 N "^-"O NOh mmM3\M>)lON^H fO •^■'■O 00 N "^-VO 00 <N "4-VO 

ro »ovo t^oo oo n nt invo oo o\0 h n -t mvo t^oo h o m -*\o t-^oo on 
m m m h m h e* w n (N m w n m rocofnmrofnforo't'+'+'^'t'+t^-'t 

On n iflO\« moN moo n moo <n moo h moo w rj-oo h ■«*■ r^ o m t-^ o rovo 

• mvo oo o nioso n *NOH tj-vo oo h ro moo o <n in t-» on n -4-vo on h ro 
t^ t^ t^oo MWwOOOO(>OOOOHHMMNNN«Nmrf)nrot* 

MM i-mmihmmmwi-.-N(NCNW«<NNP)Cn|(N(NO)<NN(NNNNN 

oooooooooooooooooooooooooooooooooooooooooooooooooooooocooooooo 



< 



fOSO "4-t^O "* t> M -<*-00 M in On N "O fOSM t*-00 NVO WNm ma 



O rovO rovo rovo rovo ro«o rot^O m f^ O mso tJ- t-^ ■* ^ O 
mton^TtTMom mvo ^oo nn r-^oo oooooNO\ONOOO | - , '- ,> -'<N(NC > iro 

NNNSNNtsNNNNNNSNtsNtsNN t^OO 0000003O00O0000000 

lommmmmmiommmmmmmmmu-jiommmmioioinirNiommm 



O 
05 






Tf00 N m O C0<O •* t^ h lO On (^!nO "^-00 NVO ^"00 PJ VO ' 



ro moo Cn) mt^o cn *no\h *<i-vo On h rovo oo h m moo o ro in t>» o n l 
t^ t^ C^OO 0O00O0 OnQnOnOnOnO m m h m CM N <N cm ro ro m (O * * •. 

romroro^rororororororO'tT(-^'<trj--t'ft't-t^-'t-t'ti-^TfJ-i 

«NCSNN«NCNlWr>lNWNNN«WWCNlNN(NNCSlNNCN)rN(NNC 



CO mvO 00 (N Tfif)NO\ w ro i/-)<0 00 N -<t-vo NOh mm NOnCJ. n ^-vo 00 



moo cm m On n vo On co<o o rONO *sh xj-oo m -<*-oo w moo cm moo (N m o 



oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 



316 



FUNCTIONS OF A ONE-DEGREE CURVE. 



h n n m « m commmco' 



■m>o«oiot 



N t^ W NNOO rO00 -^-<^^-0 m m VO N t» rO00 ■<*■ lOW t^NOO «<»• ON m ► 



5vOvOVOVOVOVOvbvOVOvbvovb>bvO 



N oo ro 0\ -*• O *0 i 



h vo o NfiOMflH NffiOM^H t*> ro o\vo n oo 



TfVO ON M ^t^O\« rj- S ON CM 1ON0 <N in N fO IO00 O rr\O00 M rOVO ON M 

O\0\0i O O O w m h m M cn <N fOrororort-Ti--^-Tfioir)Ln mvo vovo^O N 
in m invo *ovovo*0'0^o v o i o v o v o^o^o^o ( o , o^o'0\ovovovovo\o\o^o\o'0 



SOW ^J-VO 00 O N -*VO 00 H miONO\H rOVO 00 N -<J-vo ON m niOM3\N 

invo oo o\ 6 h co t}- invo tv o w w ro invo soo 6 h n ro tj-vo soo on 6* ci 
ooooooooononononononononoooooooomhhmmhm>hmn<n 

f>t^.t^t^fxt^t^t^t>.f^t^ soo oocooooooooooooooooooooooooooooooooo 



oo h -*no fovo oo h rovo on « moo h tso to inoo h -<t s on n moo h i 



ci mNO\« -<J-vo oo m roio tso cm Tj-t^o 1 - -^-vo oo o Mm saw -^-vo O h 

H H H H N N CM CS MfOrOfO-+Tj--ti-^ir)ioin mvo vo vo vo vo S S S t-»00 
rorororomrorororororororororororororororornrororororororororo 
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo» 



N "+VO 00 W rt-vo 00 N tJ-vo 00 N ^-VO 00 N t|-vo 00 N Tt-vo OO 
h m h m m w cm cm <N cm romforofO't'ttTf-'tinioioin invo 



vo m ■<*- on "*oo rooo <n s cm nnvo hvo hvo hvo mvo mvo mvo hvo cm 

ro t>« 0* ^- s 0* "4-t^M ^-oo h moo cm m o> cm vo J\ rovo o roso -^- s m* -4-oo 
ro ro ^- ■*■ ^ m m mvo vo vo t-* l-^ soo oooooONO^OOHMMCMNNrororo 
OnC><>OnC>ONC>ONOnC>OvONO v ONONO n OnONOvOn'~' 

mmmmmmmmmmmmmmmmmmmm' 



3VOVOVOVOVOVOVOVOVO 



■i VO M VO M VO I 



-ivo w s n t-s cm oo rooo ro on ■<*• on in vo i 



oo w rovo oo m rovo oo w rovo oo m rovo oo m rovo oo h rovo oo m t}-vo on m tj- 
m cm cm cm cm ro fO ro ro ■+ ^ •* >t m m io mvo vo vo vo t*» s t-* noo oooooo aa 
mmmmmmmmmmmmmmmmmmmmminmminininminmm 

CMCMCMeMOlCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCM 



o 

Q 


vo oo CM -^-vo oo N ^-vo 00 M -^vo oo cm mNf>H ro in t^ o h roms 

a -1 cm ro -*f mvo oo on m cm ■<*■ mvo soo o m cm ro rt-vo soo o o cm ro ■>*■ m 
<5-inmmmmmm mvo vo vo vo vo vo vo vo t^t^r>.t^t^r>.t^r^ soo oo oo oo oo 

NtsSNMsNNNtsrsNSNNtNNNNNSrvCstsNtsNNNNN 


s 





vo on cm vo on cm moo cm moo m ^ s» rovo CT> cm moo m ^- s o rovo on cm moo 

ro moo cm «n nomn -<*-vo' o> m rovo oo O n m s ov n ^vo o>m m moo o N 
■h- -^- -^- in m m m mvo vo vo vo s s s soo oooooooo ao>o^ao O m m 
NCNN«<NNCNNC^rNW<NNNNNNNWNNW«NNrornrorororo 
ooooooooooooooooooooooooooooooooooooooooooooooocoooooooooooooo 



O N -<*-vO 00 N -*vo 00 N tj-vO 00 N t»-VO 00 N ^VO 00 N -*fVO 00 

m m m m m w n « n Nmmnmntt'tfti'imioio mvo 



FUNCTIONS OF A ONE- DEGREE CURVE, 



317 



2 



cm rnfomfOfOtt-tT^ioiflinio mvc 






CM O in CM 00 1ON00 lONOO in CM OvO CM OVO CO ts ■* M 00 VON N <t H 

mvo * t^ m inoo CM vo O m t^ O too «h in o mvo tj- t>» m in o cm vo <n 

in mvo ^>vo tsN t^-oo oooo o o o >-> <-> ^ cm <n m m m •<*- ^t- ■<*- in mvo vo 

cs cm cm (N cm n cm cm cm cm cs cm cm mmm m mm mmmmmmmmmmmm 



W5 



m cm ovo coo s^hoo mcM o tx -<*- m ovo m w oo vo cohoo»o m h oo vo 
m *vo o cm m so m moo « -^-vo on ■<*- s o en moo « tvo o cm iAno ro 
in m m mvo vo vo s s r^ sco ooooooooooooo>-iMi-<f-icM.cMCMmm 
t^t^t^t^t^t>.c^t^t^t^t^t^r^t^t^t^t^ soo cooococooccocooocooococo 



o h mvo oo ro in t>. o cm tvo O m mvo worn moo cm in s o cm tj-vo o 

oo O w cm ro mvo soo O m cm m -<*-vo tvoo o h cm' m -4-vo soo O o' cm" en ■<*■ u , 
mvo vovovovovovovovo t^sst^t^st-s tCoo ob oo oo oo co oo MffftSa© 
ooc»c»oowooc»oooooo<»oooooooooococooooooocoooc»oooooo TO oo oo oo 

cm mt>.o cm ii-Nati ^-sa« •'S-t^OH Tt-vo o h mvo oo o co moo on* 

Oi h rovo oo cm ■«*■ h- o m envo oo cm m t*» o h *vd oo 6 en m s o cm' ■tvo" 
tj- in m in mvo vovovovo s s s soo oooooooooooooOOOO'-'mi-i 
* , t'**^***^^**^Tt-*^'t^-^Tj-^T)-^inini^ioioioir)ir, 
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 






swo\'+0vo noo -^-ovo cmoo in h scoomonco ■ 



i smo»o en ovo cm 



■«*-oo m in o cm vo o m s o * s m moo c» in o envo o **■ s *■* moo cm m o r . 
Tt * m m mvo \o« s t>»oo oooo OOOO O m m cm cm cm mmm + *i-ui 

MMMMHIiHtHMMHiHMMMHCMCMCMCMCMCMCMCMCVlCMCMCMCNCMCM 

vOvOvOvOvovOvovovOVO\OVOvO^OvOVOvOVO*OvOVOVOVOVOvOvovovO\Ovovo 






00 *0 SfO OVO N 00 VO M00 ■*• w S •<*- O NfOO SMO s •«*- N * H oo m 

w Tt-tvovcM* ■<$- t>» o" cm moo 6 mvo oo h ^t-vo o cm *no cm moo o mvo oo >- 
s s s soo oo oo o o o o O *-" m m m cm cm cm commen^^t-tin 
vovovovovovovovovovovo nnnnnnnnnnnnnnnnnnnn 
(nnnnnnnnnnnnnnnncmcmcmcmcmcmcmcmcmcmcmcmcmcmcm 



cm *vo oo o m m s o h -^-vo oo o en m s o cm tj-vo oo m i 



i m r>« o cm -*vo o 



cm m ■*• m soo oo h rot mvo oo o * cm * mvo NOO h n t mvo soo 
cm cm cm cm cm cm cm mmmmmmmm'«*-"'l-'<r'<l-'<*-ti-"<i- , <i-inininininininm 
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 



mvo oh •«■ f>» o cm ioso en moo • 



mvO O M TfVO O CM *NON 



w m moo o cm m t>» o m tj-vo oo o m m t>. o cm t*-vo oo ** mmtvo c* tt^o 
oooooooooooooo O O O m m h m m cm cm cm cm rororomfJ-** J 

ro rOforofOfOfOcorot't't't 1 t't'*ttt + 't*t'i-'t + *'*l"t . 
oooooooooooooooooooooooooooooooooooocooooooooooooooooooooooooo 



318 



FUNCTIONS OF A ONE-DEGREE CURVE. 



Q 
< 



vO ^ N OO VO ThC 



ot^iomw ooo t-^ m co n o o t-^vo -* co <n o o c^^ 



vo -+oo h 10 O m N O "^-00 « vo co t-^ w m O envo -^-oo <N vo O roh. 
r--oo oooo oo>ao m m m cn w mmmi-t i-ifl mvo ^oc n r^oo oooo a 

vOvOvovOvovovOvOvOvOvovovovovovovovovovovOvovovOvOvovOvOvOvCvo 



u-) rv><N O r^vo in co cn m ooo t^vo ■<*• co <N m o Onoo NvO 10 i- rn « m h 

In. rovo' 00 -' ■<*■ t>. covo oo h 't tsO covo a « t N covo CMN ifloo h 

m cn cn cn m romrO't'*'i- , t l 'i 1 fl mvo vovo^O nn t^oo oooooo OOOO i 

oao>o\OM>o\OM3\ooosOM>ocho\ONO>o\ot>ooaaoNO'0>c 

N N N CJ W N r *' 



CS)N(N<NNWCS<N(Ni 



Cl M N W (O n 



CN tJ- fx O CN *NO\« ifSOtN t1-C^0 N lO N N lO N O CN 1T)W CO w 

CO ^f LOVO 00 O H CO "*- ""-VO 00 ON CM CO ■<*" ID f^OO ON CN m * iO f^OO On 
COCOrocOCOCO^->4-Ti--^- T j-Tj--^-Ti-ir5LOiniOtomiO mvo vovovovovovovo r^ 
OnOnOnO^ONOnOnOvOnOnOnO>OnOnOnOvOvOnOnO>OnOvOnOvO^OnOnO s O'OnO 



n m n o^ m mm csoh m m s o h enm n ov« co in t^oo o cn Tt-vo oo 

rn i.i n o h i-ooo o n m tso h covo oo o n +m>h miooo n -3-vo _ 

00000000 OvOvOvO. ►"• H M h CN (N CN OJ cn m fO ro tn -t t M" ^r * 

in in m in in >n in invo vovovovovovovovovovovovovovovovovo'Ovovovovc 

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOjOOO 



oo mcoo t^. in cn o tv ti- cn o t^ ^- cn ONMno ooo in co m ovo •>$- cn o oo vo 

mNH inoo (N vo onto nh rt-oo cn vo o>fn (^ h LOOO 
NO vo t-» t^ c^OO oOOOOOOOO'-'HhcncncOcoco 

COcOCOCOCOCOcOCOCOCOTJ-Ti-^-^-Ti-Tj-^^-^j-^-^4. ..„_.__ 

vovovovovovovovovovovovovovovovovovovovovovovovo vo vo vo vC vo vo vo 



CO 






^vOOcOt^M ' T>N< 

t^-mm mvo vo vo t^ t 

■*" ""■ Tt" Tf Tf- ■ 



o f> in :n oocvo -^ cn o mocoh ooo vo ->*-<n ooo t^mc 



envo oo h Tt ts. o m moo w tj-vo o> cn in t^ o covo on *so covo o m <*■ r-^ 
rorocnt ^^ininm mvo vo vo vo t^ r^ t^oo oo oo oo o o o OO O - « - 

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO O^O^O^O^ONONO^ 
<NCNWNN<NNNNN«««(NC«CN<NNNN«NNWNNNWNNN 



On h <*-vO 00 M COVO 00 CO lOOO m m N CN ION N m r^ N Tft^ONCN 

^0 ^°2. 2* Q £ EP 2*" ^£> ^0° o^O cn co^-m r^oo on cm co ■*- in t^oo o O m co 

ONOOOOOOOOOOOl-'MHMMIH-IHNrNICNNCNCNCNM'OCOCO 

OO 00 OO 00 OnOnOnOnOnOnOnO^OnCnOnOnOnOnOnOnO^OOnO^OO 0*0-00 



■<t t^» o m Tt-vo oo O co m t^ o n ^vo oo O miONt>w covo oo O cn rj-vo oo O 

vooo o "Omt^ow "tvo oo O mmsoci -^-vo oo o m m n o n -^vo oo 6 co 
h >h cj cn n cn r^ mfi r'\n^^-^-^-'tminin mvo vo vo vo vc t-^ r^ r-v t-xoo oo 
in u~, in m m in m m in in m in in in in m in in in in m m m m m m m m m m in 
oooooooooooooooooooooo jaooooooooooooooooooooooooocoooooooooooo 



FUNCTIONS OF A ONE-DEGREE CURVE. 



319 






ON O\00 OOOO SN t^vo vovovovcvovovovovovovovovovovovovo I 



< 



x 



u 



O 01 VO -*-0O N^O ^"OO N^O O ^00 CHO O ^00 N« O ^00 N^O "4-00 
H h CI N N fOM + ^tifl IOVO VO^O N t^00 OOOOOnOOOO^mCNNW 

t^t>.t^t^r^r^t^i>.r>.t^f^r^t^f^t^r^r^r^r^t^t^r^ t-^co oo xoooooooooo 
vOvOvovOvOvOvOvOvovOvOvOvOvOvovOvOvOvovOvOvOvOvQvovovovOvGvovo 



<^0\OOM>0>0 o i 



M w re re -<f -^- to mvo t» t^oo O m m « re 



n moo •- ■* r»» h -<f t>. revo O oi moo h ^-nq revo o> N moo N inoo w -^■ 

Ov o\ o m m m n o) n 0) romn^t^inmm mvO C^O NN f^OO 00 

000MM««MHHMIHMWMWMl-.MHMMMMWMMHh-WH 1 

corocncofOforomcnMfnfOcorofnforoforocomrocofomfnnmronro 



t^ O m iflNO re moo w tj-vo ON ■* t-» re moo w "i-vo o 01 m t^ revo oo 



oo O- N ro mvO t^oo O w (N re mvo NO\0 h « t mvo saO m re ■«*- mvo 

OOi-HHMHM-NNOINNNNNrOrqrOfimrfimrO'f't'tt+'t 

ooooooooooooooooooooooooooooooo 

CSNNNNNNNNWWNCSNNWNINClNOlNNNrvlMCNNOiNW 



re ^j-vo t^ O c 



■«*■ m in. o cj re mo oo oh re -<4- m t^co h re -^-vo t>* o 



■^■vo ooOoimr^OHro moo 01 -rvo oo o re m r>N o h reo oo oj -^-vo oo 
m m m n w cm n w rererere-^--^-^-^--^-mmmio mvo vo vo vo t^t^o-t^r^ 

oocoooocoooooooooooocooooocooocooooooocooooooooocooococooooocc 



7i 






vo m -^ <n m o oco c^vo m -^- re N h o Oco t*s r^vo m-^-Tj-reNNMMMO 
m to o re t^ h tj-co nvo o tj-co mvo msh mo-rer^- m o re r^ h mo 
o o o h h m n n rerere-^'^-mm mvo vo vo t>^ t^oo oo oo aao 

m m mvo vovovovovovovovovovovovovovovovovovovovovovovovo r^r^t^ 
vovovovovovovovovovovovovovovovovovovovovovovovovovovovovovovo 



OCO CO t>.vo vo m m -* re re re N 0) h h h o o o^o^o^o^o^a^c^o^c^ 
■<4-vo o cs moo w ^- r>» o revo o cm moo m 4- t% 6 revo oo h ■* >- o" revo o oi 
Q 2 2 2 2 £ £ £ ^ST'SP^;?"^"^" 1 ^! ""^ ^^^ nn «^oo oo oo oo on 

~ooooooooo~ 



oo revo 00 w revo O m Tf vo O N tJ- ^ o N to tv re moo h revo o\h ^s 

oi re ■<*• m t^.oo o oj re rt- m t^co o f oi re -^-vo t^oo on h oi re tJ-vo' t^.co 
r-» r-» t^ r^ t-s t^ t>» t^oo oooocooooooooo ONOOoaooo^o O O o 

OMX^ONOM^OiOO^OMjiOOi^O^O^O^O^OvOO^OO o o o o o o 

MHMMMHHHMHWMMMHMMHMWMMWMNWNNNfNN 

04 re m t^ ov h re-*vooo o n remr^ovO n ^-vo t^ on m. n -«*-vo oo oh m 

O m re m f^ O N -^-vo oo re m r^ O « revo oo O CJ "*-vo O-w remr^ooi -* 
-<4- m m m m mvo vo vo vo t-^ t>. t-^ r-~ t^oo cocooo oooooo o o o o m m 

vovovovovovovovovovovovovovovovovovoovovovovovo NNNnnns 
oooooooooocooococooooooocooooocooocooocooooocooooooooocococooo 



320 



FUNCTIONS OF A ONE-DEGREE CURVE. 



< 



Q 
« 

o 

6 
& 



o 

u 



O n m -^-^o t^oo o n cm "*-no r>. o> o cm co m t^ on o cm ^fv© co o cm -*vo oo o 

h 10 o* ro n h mo ■'foo cm vo Q ■<*■ On co t^ h u-> o> -too cm no o mo\rosH\o 
m 10 mvo no t^ t — cx3 ooooooOOOMMNNwrom-<*-'^-ioio u-ivo vo ns 
OnOnOnOnOOnOnOnOnOnOnOnOOOOOOOOOOOOOOOOOOO 
nonovononononOVONOnononO t^C^t^t>t^t>t^t^f^r^t^t^t^t^t^t^t^f%t>. 



co m t^ On 

00 H +Nh tN 



tomso»H coior^OH -<j-vo oo m inoo o co moo m t-* o i 



_ co t^« cono On cono onvo on in o cm moo cm moo cm in 
t^oo ooooONOOOOO'-'WHHNMwrororoT^ThTj-iom iono \o>o nn 
cm cm cm cm n cm cm ronfOfomrofOfOMronforfifncnrommrommmmm 



CONO 00 h *S0 cono On CM -«- f-* cono On CM inoo h -«-no On CM inoo h ^- f* 
u->vo f> On O m CO «*■ iono O0 On (M co "*■ in r^oo o H CM CO "^-VO t^.00 Q M N » 

coooooooo\o\»o>aoooooooooooHMMHHhHNN«N 

OOOOOOOOOOOOHWHHHHHWMI-l>HI-iH>-MI-lMMi- 

CMCMCMCMCMNeMCMCMNCMCMCMCMCMeMCMCMCMCMCMCMCMCMCMWCMCMCMCMCM 



00 On CM fO ■* lOVO 00 ON H CM rOT|-inN00 On i 



N fO't lONO C^OO O l 



cm •^■t^OM mm nonh -^-md oo o n ttvo co o rnifiNOM rotnr^o>M TfNO 

Tj--t-ti-lOlOlOl^ lONO VO NO VO t>- !>. t^- t-~ t^OO OOOOCOCO OnOO^OnOO 
COCOOOOOOOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOON 
CXJCOCOOOOOOOOOCOCOOOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 



© 
© 



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o 
s 
u 



000000 On On H h CM N CO ■«*- ■«*■ lONO t^OO 00 On O h CM CO •«*• lOVO t>.00 On 

OO* CM* NO* ^ONfONH lOOfASH in On CO t*» m in -<*-oo CM VO "<*-CO CM NO H 
cm co c» ■*■ ■<*■ -t- io iomd no no r-^ t^co oooo ooO h h m cm cm co co co ■<*•■*• in 

OOOOOOOOOOOOOOOOCOOOOOOOOOOOOOOOOOOO OOOOOOnOnOnO^OO^OnOn 
VOVOVOVONONONOVONONOVOVOVOVOVOVONONOVONONOVOVOVONONONOVONONOVO 



CO •*■ IOVO t^CO On h cm ■*• »OVO NOO w CO ■«- lO t>.00 CM CO lO tvCO CM CO 

•4- t^. d cono on cm vd 6- cm inoo h -4- t^. m •«- t-* d cnvo on cono o cm inoo cm inoo 
CO oo On So> ONOOOHHHCMCMCMcococo-*-T*--*Tj-ioin invp »«nn k 
v^ yy .y y. y y. ~ is x x ^ M M M N N cm cm cm cm cm 
■jcococococococo 



00 H •* fv On CM IOOO M CONO On CM lO «>. CONO On H "J" t-» CONO On H "«f ^» CO 
vo'oo OnO M CO^lO t^OO ON cm' CO 4-no t^co O M CM CO lONO NOO m W *• lO 

^-^- ^-ioioioioioioio lovo vo no no vo no no no JL- t> J^- 5> J:- J> ^°S °R °S °2 °S 
oonoooooooooooooooooooooooooooo 

CMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCNWNCMCMCMCMCMCMCM 



On h CO ^NO r^ On On co ■* »o t^OO O m CM CO lONO t^OO h i cs I CO -^-no t^co 
oo h co io r^. On m cono' oo 6 cm -<*-no' oo ro m N On « co iooo cm **no oo cm 

tCoO COOOOOOOONONO>ONOOOOOMMM-MCMCMCMCMCOcncO ™£r-*t + 
t^^JtLt^t^. tC t^ t>. t^OO 00COO00000000000000000000000000000000000 

cooooooooooooooooooocooocooooooooooooooooooooooooocooooooooooo 



O CM ThNO 00 O CM -*NO OO O CM ^NO 00 CM -<*-NO 00 O CM -<*-NO 00 O CM "*-VO OO 
wv. -.-w WMHlHM CMNCMCMCMCOCOCOCOCO^-'"*-'^-^'<t-U-)lOiniO lONO 



FUNCTIONS OF A ONE-DEGREE CURVE. 



321 



© 



Q 
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vo o n moo i 



i tooo N m On COVO O "<*• t>. M m o> ro t^. ► 



ro n nvo O m o> rooo cm no m m on -<f oo cm t^M mo * o> rr- n ei \o mo> 
OOwHNNCJmm^-^mu-) mvo vo t^ t^oo ooooooOi-iwnnn 

CMfNCMCMCMCMNCMCMNCMCMCMCMNCMCMCMCMCMCMNCNim rommrOrOrOrO 



VO ■*■ t»» W m 0> mvO CO -*00 N l^» H U"> O m f^ N VO »n On rOOO N NNVO H 



">4-O0 m -^-OO H tTCO M inOO M IDOO CM lOOO CM m On CM VO ON CM VO O COVO fON 

t>» t^oo oooo On o> On w "- 1 •- 1 cm Pi cm romro't'^'+inio mvo <0 nsn 



mvO O" CM i^CO M tNH ■<*• (^ rr)\Q On CM moo 



■<*• t-» mvo On ro 



CMCMCMCMCMCMCMCMCMCMCMCNCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCM 



t^OO O^ On ' 



N fOM'tm mvo tv t^oo OnOnO m cm cm ro m ■<*- m invo no 



oo cm *nc^h roioN^H fOmNONH re moo cm "4-vo oo cm "4-vo oo 

OnOnOnOnO>OnO>OnOnOnOnO>OnOnOnOnO o o o o o o o o o o o o o _ 
00000000000000000000000000000000 OiO>0>OnO\OnOnONOiOnO>0>OnO>On 



o 



Q 

< 



<s cs -cm cm cm ro ro ro ro co ^ 



- •* -^- m m m m mvc 



CM -*VO 00 H miflNO CM M- t^ On CM n+NO CM 1ON0 fOVO O h ■<*• S mvc 

vo o -<i-oo cm t*» m m on -^-oo cm no o m o> moo cm vo" o m o> m tA cm no' o" m o* rr> 
t^oo oooo ononO w m cm <n rororo-^Tj-in mvo nQno s t^oo oo on on On 

OOOOOOMMHMMMMHt-IMHMHi-IMMMMMlHMWWMt- 



h mvo on ionO mvo On <m m On n moo m moo m -^-co w moo cni m on cm vo 

mOO M ^00 H "*00 M Tht^lH -tNH ^-t^H Tf t^ H "* t^. M -^t^M tt\H <i 

t^ t^OO OOOO OnOnOnO " m M N CM CNI CO CO CO <t t ■+ lO m ir>VO ^OVO NN 
mmmmmmmm^*^*Tt-'<<--<*-Tj--*-«*-Ti-*Ti-Tj-<«-Tj-T}--<t-Tj-Ti-^-Tt-.«- 

rocomMcorofomrocofOforommcomrocnroromrommforomromfo 



mvo Os N moo h •<*- t^ o mvo o cm moo m +no mvo on cni moo h Tfso 



CMWCMCMCMNWCMCMCMttCMCMCMCMCMCMCMCMCMCMeMCMCMCMCMCMCMCMCMCM 



■I CM m ■*• mvO t^OO On I 

j oo d cm -*vd oo o' i 



N co* mvo vo t^oo o i 



' N rotio mvo t^ 



322 



FUNCTIONS OF A ONE-DEGREE CURVE. 






lOO >00 lOO lOH^O H^O N S rOOO MO'tO^O H s roco -<J- VO N CO *<*■ 

NMU) h ir> 6 i-a roco N i 
vo S SCO CO o> Oi O h i 

sssssssssssssssssssssssssssssss 



»0 
© 






vo h s co o -<i- vo woo M- O vo rnOL 



Si- O^O fO 0*0 IN O to c 



NVO 0\ fOvO O "*■ S H ^00 tN tOOCJMD o CO S i- 

oooooo^aooOHHHNNNnnt'ttu 

vo vo vo vo vo sssssssssssssssssi^r^sssssss 
forocoromtororocorrtOfOfornfototoforotocororofOfotototorofoto 



co h *too m t}- s w <<*- s ■<*- s MNO (O s tovo covo covo o covo 

h n-tm SCO o - pi to tovo S 0> h f^ -<*■ to SCO o h n co tOMD SCO o h 
't'tti-'t-'t'tiO"TniOiniO tovo vo >o vo vo vo vo <o S S S S S S SCO CO 



X 
O 



COOOnOOOOOOi 



N N N N N COCOfOCOCOCOCOCO fO ■<*•-« 



H CC) m S O IN tj-VO CO N ""^VO CO IN Tt-vO CO N ^VO CO N rt-VO CO N 

OOOOOnO m pi — m ii cn in cn cm cn roroMnron-^-'ttmin 

OOOOOW-MHMMM-^-MMMMMHMMMI-.MMHWH^ 

O^0~O\O^O^0^0^0^0^0\O^0^O^0s<j\0\O o s o^o^o^o^ovo^oo^o^o^O N o^o* 



z 

s 



Q 



H to ON CO S H lOO "<*-CO N VO M lO "<i-CO (OCO N NHVO H VO "1 lO 

1-CO N S N VO O ■* Ov (OCO (N S >- VO O rj- O (OCO N S M VO 6 lO 0\ tK30 CO S 
CO (O ^- Th to tOVO VO MD S SCO CO OOOO m m n (N ro fO 1- rf tJ- in tovo vo 

sssssssssssssssssssssssssssssss 



o 






m lOO "10 >tO>'*0>'tO'tO^(>*0 tOO tOwVO t-i S N CO CO ON •«*- vO 

so ism tj-sh 1-co h u-,co n mo* (ovo (O s o' ■«• s w i-co n m o f) 

SCO COCO OOOO m h m N CN N (0(0-!t-Tj-Tj-u-)in mvo VO vo S S SCO 
lO to to IO to ID invo vovovovOvovOvOvovovcvovovovOvovOvOvovovOvOVOvO 
cocococococococococococococococococococomcorocococococococop-: 



(OVO ON N »OCO N lOCO M •*- S ■* S C^VO (OVO ONNVO (> « lOCO N lOCO 

CN (O tJ-vO SCO w N "^ tOVO CO O N (O •^"O SCO O M N (O tOVO S O ►» 
000000HMM«MWi-iMNMN(NPJN(MN(Om(O(O(O(OfO"+-<f 
NNN(NNWC4NN«NtN(NCSMNCMC\WNNNNNNNNCMCNNC» 

NNNNN«<NNN«NN(INNN«NN«NNCIBNNWNCINM 



vo S SCO 00 0\00 h h 



IN n ro (O (O •<*• - 



to m tovo vo vo s s sco co co 



N 1-vO CO N ""NO" CO to S O w (O "O S ON ~ "O m S ON ~ (0"-SOv- 

ro ro c " n + + + t t m m m in mvo vo >o vo vo ssss sco co co co co c 
ooooooooooooooooooooooooooooooc 

C^OlC^C^C^O^C^C^O>C^OlC^C^O^C^O^C^C^O x C^C^C^O^C^O^C^C^O^C^O\C 



FUNCTIONS OF A ONE-DEGREE CURVE. 



323 






t^-^-MOO lOrOO MON MOM S^fOHOOvO "* N 00 SO ■*• CM Ot>. 

COCX) rlNN ts nvo mVO H in lOO -<*- Os -<t- Os <"O00 rOOO r^NN t^ cm t^-vo 
•<f ^- m iovo o t^ t^oo cx3a>ooo>-<MH(NNromTj--^io m\o so t-> r^oo oo 

t^t^t^r^t^t^t^t^t^t^t^ t^OO OOOOOOOOOOOOOOOOOOOOOOCOOOOOOOOOOOOO 



H 0\MO(^H On t^vo -"4-N OMOiJ-N H O00 SO lO^fOH O00 t^sO u"> 

COsO -<1-00 CM mO\mNH IDOO NO "^-00 m lOCfONH ir> 0> CM SO ^00 
OOwMMMNCsrOrO-^-^Tl-io IDO ** NN t^OO 00 Os O O^ m m m 



t*. ^-N0 ■* t"» ► 



■* t^ M rj-0O I 



IO00 N i/"> Os CM so O MVO fONO 



m fOiflO t^OO OiH N fl \T)\0 NO\0 hi CO "tf- U"> t- 00 O M N CO tOSO 00 OO N 
CM CM CM CM CM CM CM rOrOfOfOrinm't'tft^'t't'tlOIOlOlOlOlO 100 SO 

CMCMCMCMCMCMCMCMCMCMCMCMCSCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCM CM N 



u 



CMNCMhmhmhhOOOOOsOs O00 COCO SN f^so so so m m ■«■ tJ- •*■ m 
CM -<i-so oo O N -«-vo' oo 0* CM "^-mt^OHi romst>M romsaH miosom 

M M M M CM CM CM 04 <N fOmrOfOCOrO^TfTj-Tj-TflOLOtOlO l^VO VO *C O) VO N 
NCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCM 
C^OsOsC^OsosOsOsOsOsOsOsOsOsOsC^OsOsOsOsOsOsOsO^OsOsOsOsOsOsOs 






SO MOO ^0 NfOOMONCO iflMOO ->i-Hi l^^-M N^-HOO IOHCO i/> N c r«x 



o 



w oso fO Osso no NtH o<o mo MAS t^-^-w o Nm« Ooovo CO h 

hi -^-oo cm iooiwnO "+00 h ioo ro*0 "i-oo m \r> o coso -^-oo' h! iA os ro 
Os Os os o hi hi cm cm (N mcnro-tTtiom ioso CO n f^oo oo oo ONO\ O 
t^ t^ t^oo oooocooooooooooooooooooooooooooooooooooooooooooooooo <-s 

romtocororomrocotfimmfOcOfOmtorocnmcorOfOrofOrofOrororon 



so 0> COsO Os roo o mso Os fO\Q Os rovO Os rovO Os rovO fOO fO N o ns 
hi <n ■<*■ LOVO 00 0> CM rn -*-sO t^OO hi CM ■<*• irsso 00 Os hi n ro io<0 t^. O- m 
OOOOOOOOOOOOOOO^OsOsOOOsOsOOOOOOOOhmmmmmmnm 

cm cm cm cm cm cm cm cm cm cm cm cm cm cm nmmmmnmcommmrommfnroro 

CMCMCMCMCMCMCMNCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCNWCNCMCMCM 



^•^Tj--*'*^-^--" 



"**^^ , + + n«.cnwconc 



CM tJ-SO 00 CM 

O HI M 
IHMIHMWIHMI-IMIHI-IMMM-IH-l-IMMMt-l-lfc-CMCMCMCMCMCMCJ 

OOOOOOOOOOsOOOOOO' OsOOO^OsO O^OsOsOsOsOsOsO^Os 



324 



FUNCTIONS OF A ONE-DEGREE CURVE. 



o 

© 

© 

H 




N -^-VO 00 O CM -*VO 00 O N -«-VO 00 O « -^VO 00 N "*-vO 00 CM "*VO 00 
H m h m h CM W CM W CM COCOCOcO(OtJ-^-<j-t1--«-\/->ioiOiO u->vo 


< 


CM M H O On O\00 000000000000000000000000 O O OS M H CM cm co 


cooo moo co r^. cm r^ N t*»cs t — cm t>. cm r-^ cm t^ cm f«s cm t^ cm r-^ cooo rooo cooo co 
O ro -<*- "=*• m u->vo vo t^ t^OO 00 O\C>0 h m CM cm ro ro -<i- Tt- io iovo VO I s * t^OO 

OOOOOOOOQOOOOOHHHMMHHHHHHHHHHMH 
OOOOOOOOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 


u 

•w 


't't'+lfllO IOVO VO t^.00 OOOOOHCMCO"*- lOVO t^OO O H ro rj- lO tvOO 


t^.M looroNH 10 o> co t^ h vo o f^oo cm vo o "*-oo cm vo m moroNH ino 
ro -^- ■>*- •* u-j u~,vo vo vo t^ tN.oo oo on O c^ m m r h cm cm co ro ro t|- ^- io iovo 

MMWHHHHHHH«HHHHHH CM CMCMCMCMCM<NCMCMeMeMeMCMCMCM 


Q 

o 

Q 


vo o N > covo ^t- tN. m Tt-00 N lO On CM VO rot^.0 "*• h >0 0\ CM vo rONM M- 


CM CO IOVO 00 ON <N ro ^-vo t^oo H m Th ID t^OO H N CO lOVO oo o o cm co 
OOOOOOt-iHMMHHHNWNNNNNfnromrororomro^Tt-Ti- 

CMCMCMCMCMCMCMCMCMNCMCMCMCMCMCMWeMCMCMCMCMNCMCMCMCMCMCMCMCM 


Q 

O 
X 

u 


oo t>«vo ^lottms « h oo\ onoo t^vo voio^cocmcmhO o>oo t^. txvo m 


Oih fOms(>H romNan cm tj-vo oo cm -*-vo oo n ^^ no\m rcms 
cm fomfOfOfo^-'t'tTj-^iominin iovo vovo vovo r^» t^ t^ r>. t-» r»-oo oo oo oo 
ro m m tn n ro co ro.ro ronnfomromforofotnrofomfomnrnfonforo 
0N0^0n^0n0N0vOvQv^0\0n0nOn0^0>0nC^O>0^O>O\O>0v0\0>0v0nOi0n0n 


2 

2 


CM "*VO 00 CM ^-VO 00 CM Tt-VO 00 CM tJ-VO 00 CM -<*-vO 00 CM M"VO 00 
M M M ft H M CM CM CM CM COCOfOCOrOT}-'<*-Tj-Tj--^-lOlOJO»0 IOVO 


o* 

00 

© 
H 




CM -^-VO 000 N "♦'VO 00 CM -^-VO 00 CM -*VO 00 « "*VO 00 N -^-VO 00 
H M M M H W N N CM CN COCOCOCOCO-^--^--q-Tl--^-u-)u-)iou-) iovo 


A* 


t^lOCOCM 000 Mfl'tOH 000 t^VO lfl'trOH OV00 00 t^VO lO rt- >^- CO CN N 


VO HVO HVO «O0 ^0 1O0 M-ON'^-0\-<d-0\'^-ON rooo COOO CO00 CO00 CO00 CO 

00 O>On0 O m h eg N cocOTj--<*--<ihu-) iovo vo f^ t^OO OOOnCJnOOhmnwco 
000000 0\0>0>0\OvO>OiO\ONOiONOOM>0\f>OONOOO>0 

c^cvt^t^t^f^f^cv.t^t^tvtv t^t^t>.t^t>»t^f^t^t^r**f^ r^oo oo oo oo oo oo oo 


u 

w 

X* 


io^con h ooo oo r^vo vo uoioiO't-t'+'+rococc. cocoforororo'*'* 


oo cm 'O O "*-oo (N io a ro n h ioo\rosH io r>roSH ioo\conh ioO\con 
m n CN rorrirn'^--^--^-iO iovo vo vo t^ t^OO 0000O0>000Mi-iNWCMC0m 
OOOOOOOOOOOOOOOOOOOOOwhmhmhmhmm 


Q 
M 
O 

Q 

S 


CONO * NH tJ-00 H »O00 N lOfJ COVO CO N ^ t>. H -^-OO N I00«\0 


cm co Tt-vo r^OO "h CM rj- iovo 00 O N co iovo t-x O h ro ■<*- in r^oo Oh ci 
vovovovovovo r^r^t^r^t^t^r>» r>.oo CO oo co oo co oo o\0*f>OOf>0>0>0 O 
rorocococorococococococororococorocorororocococorococococo't^' 

CMWNCMNNNCMNCNWNCNNWNWMNCNINCMCMWCMrMMN 


Q 
% 
O 
S 

u 


COCOfONNHMOOON ON00 00 t^ t^v© «lfllO + *fONNH H orj\ 000 OO 


• m roiONOH roio r^oo n ^vo oo cm ^fvo oo cm -^-vo oo O cm romNO* 

t>» t^ t>» t-> C^OO OOOOOOOOCM^O\OiaOOOOOHHHHHNNNNMN 

cm cm cm cm cm cm CM Cm CM cm cm cm cm cm cm rocococorocorocococorocorororoco 
O^0^0^0^0^O^O^O^O^0^O^O^O^0 N 0^O^0^O^a^0^O^0^0^0^0v0^0^0^0^O'0 N 




O CM -^VO 00 CM -^-VO 00 N -^VO 00 CM -*VO 00 CM i*-VO 00 N "♦VO 00 O 

u cn t>u wu ^ ^^^ £$ N N N N rrrororof, *^*^'tioioioin mvo 



FUNCTIONS OF A ONE-DEGREE CURVE. 



325 



Q 



Q 
K 

O 





u 



n N « n N nroffncO' 



N ♦VO 00 N mNO\M -«-vO Ov H -*\0 ON N ' 



• t^ o mvo o « moo i 



sn s«oo moo coco i}- o> ■<*■ omo o in o vo mvo w t-^pj t^ moo moctaio 
rrt^-io mvo vo t>» t^oo oo onO\0 h m m n m m ♦ ♦ m mvo vo t^ t>«oo oo o> 

oooooooooooooooooooooooooooooooooooooooocooooooooooooooooooooo 



*N0 MVO 0\ N moO m mOO 



li"500 N U")Ove«vO fON^ 



lOO>fONH 



vo o loorosNO o m o> moo wo m mo ♦oo m t^ m vo o ♦ o> moo n vo 
oo o> ov o> O m m m w n nfit + iom mvo vo t-* t^oo oo o>o\oo O M M 
mmmm^^^^^^^^^^^^^^^^^^^^^^^mmmm 



mo^Nvo o ♦ t^ m movwvo o ♦oo h ioo mvo •♦oo n in on m t^ m moo 

♦ m f^oo o h ei + mvo oo ov h n m mvo r*» on o n m ♦vo t^oo « m ♦ m 
oooocooooo\0\00\0\(JO\OOOOOOOMHHMHMMnNNCiti 

♦ ♦'♦'♦-♦•♦-^■■♦♦•■♦•♦♦mmmmioii-iLniommmmmmmmmmu-j 



m ♦ m m o onoo t^vo ♦ m n h o oo t^vo in m n w o oo nvo ♦ m n o onoo 

♦vo oo O n mmt^ovH ro io s o (i n ♦o oo 6 w ♦ in t^ o> m mm t^oo Q 
•^-♦♦loiommm mvo vovovovo i-» t^ (-* t>» t^oo oocooooooo o>o\Ovo>o>0 
♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦^♦♦♦♦♦♦♦♦♦♦♦♦♦♦m 

ONOnO>OnOvOvO , O v O>OvOvO>0>0*0 > iO\0\O x O>0>0\OiO > iOnOvO^O>0\Ov0^0 1 i 



Q 



m ♦ ♦ mvo t^oo on o -i n m ♦ mvo ooao n n mvo oo o\ m m mvo oo o n 



moo moo moo moo ♦ o\^o\^on^o>^o mo mo in o vo mvo mvo n , - 
oooo ovO\0 h m n n m m ♦ ♦ in mvo t^ i^oo oooNOvOOMMNpjmm 
HMMMCMN«NCV](N<NcsN<NNcsi(NcsMcv)N(NNmmmmmmmm 
oooooooooooooooooooooooooocooooooooooooocooooooooooooooooooooo 



i m ♦vo oo ovm mm^ovM mmt^ovi-t ♦vo oo m mvo oo m mvo ov w 



i n w w « n n n mmmmmmmmmmmmmmmmmmmmm 



«j-oo m ioohvo o mt^M -^-oo n vo amso ^00 h 10 Ov mvo ♦ t*» m m 

m ♦vo t-^00 o m m ♦ m t-»oo o> m n m mvo 00 on o n m ♦vo no\6 h m ♦ 
♦ ♦♦♦♦mmmmmiom mvo vovovovovovo t^t^t^t^t^.t>« t^oo 00 00 00 
♦-♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦■«-■«-♦■<♦■«-■«-♦'«■■«■-*■•«■ 

«NCV1W«NNNMN<NNNNNMN«NMCMNW«WWNNNWW 



onoo t^ t^vo m ♦ m n 1 



C*oo nvo m m n 1 



000 t**vo m 



fv on m m in t^oo o n ♦vo 00 o n ♦o 00 ovm mmt^o>M m mvo 00 w ♦ 
ooooovovovovOvoooooMMMMMMNNNONm.mmmm^^r^ 
(^(^(^{•^(^{^(^♦♦♦^♦♦♦^-♦♦♦♦'^"^■♦♦♦•^■«--«*--«i--«-'<*-<<t- 

OvOOvOnONONOvONOvOnOvOvOvOvOvOnOnONONOvOvOvOvOvOvOvOvO^OvOSoS 



326 



FUNCTIONS OF A ONE-DEGREE CURVE. 



o 
CO 

H 
H 




N -^-VO 00 N -*vO 00 CV1 -^-VO 00 CN -<l-vO 00 N ^VO 00 N "*-VO 00 


d 


h*oo mO>oO "iO mO^o h\o « n cooo ^-omomvo « oo "<t- vo woo ■*- 


tsNoo co o\ •*- m h vo w t>« cooo ^-omoOvo w r^ cooo ^-0\m mvo w t»ro 
vovo vo t^ t^oo 0\00 h m « N rorrtm invo vo t-> r-vOO 00 O — m N 
vqvovovovovovovo t^t>.t^t>.t^^«t^tr^r>«t^t^t^r^t^r»t^t^ t^oo oo oo oo oo 
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 


u 

W 
C/J 

X* 


VO NOO +OMOH NmOMOHOO ■<*- VO co owo N OMAN owo N OWO too N 


M VO <0 0\*0\ COOO N NNvO MVO uiontJ-O COOO CO tv W ts h VO M vo 

io mvo vo vo t^ t^oo oo coo o m m « n n nnt^if) mvo vo t^ t^.00 oo Ov 

vO VO VO VO VO VO VO VO VO VO VO NNNNNNNNNNNNNNNNNtststv 


Q 

o 

Q 


■^-00 N VO -C-00 H i/>0\«Nh V) OV CO tv h mO^mSH '^•00 MVO o -^-oo N 


t^oo h co -+■ m t^oo Ov m n -^ ir>vo oo Ov w n co mvo oo O^ N co mvo b» o 
VOVO NNNtsNtsN t^oo OO CO OC OO 00 00 OvOvOOvOvO^OO 
ioinminiflioioioioioioinioini.Tinioinmioi^ioin mvo vovovo^o vovo 

NNWNNN«NNNM<NNNWWN<V)NC*CSNCVINWCV1C1(NCSWW 


Q 

K 
O 

X 

U 


co n ooo tv u-> ■<*- n o Ov t^ "■>•«- « ooo Mnmn ooovo t^h Ov t->. m> ii- « 


VO00 w fOinNONM N ^t-vo oo w co m t^ Ov w co tJ-vO oo N mmNO>M 
in mvo vovovovovo t^t^t^t^ t^oo oooooooooo oovoaoo I" 4 
ioioioio>oinininioioir)ir, mioioioioioioiriminin mvo vo vo vo vo vo vo 
OvOOvOvOvOvOvOvO\OvOvOvOvOvOvO^OvOOvOvO\OvOvO^O>OvO>OvO^O<0^ 




N "*VO 00 N i»-vO 00 « ^vo 00 N '■a-vo 00 d tJ-vo 00 N -^"VO 00 
m m m m h « cs o* cs (N cococococO'<*-'<J-Tt-'<a-Tj-miniom mvo 


o 

H 
H 


55 


N ^VO 00 N "*VO 00 N -*VO 00 W -*-VO 00 W "*VO 00 M "*-VO 00 

MMMMHNNNdcicotorocotot't'J-ttininiflio mvo 


d 


M tJ-00 M -Tf-OO M xn Ov N VO O -^-00 N VO -<J-00 N VO m OV COOO B NN^O H 

io6 iom'vo m r»» ci *>» cooo -<1- Ov ->»- O in h vo m t^csoo rooo 4- Ov m vd m t^ 
OvOOHHWNrorooj-^ir) mvo r- r^oo oo C^O\0 O h m n n mttiom 

^•mi01/)lT)10>0101010UTT''lTOlO'010'0 IOVO VOVCVOVOVOOvOvOvOVO 


u 
w 
c/J 

X* 

W 


Ov -^-00 N VO M if)0 "<*- Ov COOO fOOO fOt^.N NCI N N CO COOO fO Ov * O «0 H VO 


VO H iTi ■*■ Ov COOO MVO - iO0 ■*- Ov COOO <N Nw VO '-O Ov ^00 COOO « f^ »- 
H N N CO CO CO ■*-■<•- IO l^vo ^O NN t^OO OOO^OvOOMiHHNNCOCO^-^lO 

mmmW)Uim xr > ir > xr > u ~> ir > l '~> ir > ir > ir > ir > xr > l/ ~> "^vo vovovovovovovovovovovo 


o 

Q 
S 


00 NVO +NH IO OV f O tx "^-00 NVO oVt^w iOOvWNm lOOO NOO* 


io r>»co o m cs ■<*- mvo oo o> h w co mvo oo ao n co -<»-vo saO h co <<j-vo t^ 
N o N mrOfOfOfOrorrfO'*'+'l-' 4 -* , * , ' | * ,1 oiOi^iCi'^>n irjvo vo vo "O vo vo 

WMCiNWC^NCSWNNNWNNWWNCSNWWCSWNWCSWWCSN 


Q 
g 
C 
K 
U 


00 vo IO tJ- M M OS00 MO^rN h ovoo vo IO CO N Ov t^vo -^COM ooovo mco 


N -^-VO 00 " miONO>H CO -^VO 00 N -^"VO N Ov H « IO N O N -*vo 

OOOOOMHHMMHNNNNNforofiromrottt'ttioioioio 

^OvOvOvO>OvO>OvOvOvOvOvOvOvOvOvOvOvOvOvOvO>OvO>OvOvOvOvOvOvOv 


2 


« '^' , 00 C W -*vo 00 N "<*-VO 00 N -^-VO 00 N -*VO 00 N -«-VO 00 Q 
H M M H M 0) N CM N N COCOCOCOCO'^-'^-it'^'<4-lOlO»OlO IOVO 



FUNCTIONS OF A ONE-DEGREE CURVE. 



327 



O « -<t-NO 00 O N 



i ovo ■<*- n Ooo t^ vo ro m ooo surtN i 



oo f*-NO io*roN h h O 



.. ro ^- tJ- lo iono no t^oo co oo^O 
0000000000000000000'->-i>-'--m-mn-i«mm 
00 OOOOOOOnOnOnOnOnOOnOnOnOnOnOnOnOnOnOnOnOOOOnOOnOn 



ON00 t^vO vo "*■ ro ro 0) N 



h o 0000000000 



-<t- O Tf On "too rooo rooo rooo rooo rooo rooo rooo Cl t^N t^-IN t^-N ts M 00 rO 
roro^^m iovo no t» t^oo oooooOw»-<NNrorO'<*-'<*-vo vono no t^ t^oo 
O0\O0>OO0\O0\OOO0n0n0000 0000 000000 00 



Q 
PS 

O 
d 


msM ioo-fOSH-mo\ ^-oo n no "*oo n no m o^ n s « vo o ro t^ n no 


m n t iono oo o>« n f) vono oo o w r» ro iono oo o m ro mvo t^ o n ro 
m io m m io in IONO vovovovo^o^o nnnnms t-^oo oooooooooooo ooo 

vOnONOnonOnOnOnOnononONONONOnONOnonONOnOnonOnononO^ONOnOnONOnO 


Q 
K 

o 
s 
u 


comoovo ^- « ooovo ^-h (OMomOoo^O -<*• w o t^ io n oo no m« ono 

»n r^oo O n ■<i- v O NOm m -^VO OO N f^iONOO n -^-no oo O m ro VONO 00 
vcvo\o r^t^r^r^t^ t^oo oowooooooooooooOOOCh- m m « 

vONOnOnOnOnonOnOnOnOnononOOnonononONOnO t^t^t^t^r^t^f^r^t^t^t^ 

OOOOOOOvOOOOOOOOOOOOOOOOOOO^OOOOO 



u 



■^■M NfOO^O N OlON OMAN OVO fO S'tHOOO ro 00 lfl CO 00 mm 
fOO *0VO H ISMOO i-OlOH^O (N 00 "^- O VO w NO N 00 ^- O VO m NNOO •**• 

<n n ro -^- ■>*- io iono no t^ t-^oo OOO «-■ h oj ro ro t}- tj- io iono t^ r^oo oo o 
oooooooooooooooooooooooooooo ooooooooooooooooo 
oooooooooooooooooooooooooooooooooooocooooooooooooooooooooooooo 



MON OnO "*■ M OnvO +« ON*N 00 NO t}- C* OMAMN O^ t^NO ^- 
vo -«f- O ■*• On rooo rooO N f^N N M ^O n (O H'O o'lOOiCC vo On tJ- O "i- 

9 s P^-S ,5 .5 J2 _~ .£! .£ -<r > JT 1 J?'J3"J£ 1 Jf>bf5 ^? *> ^9° ooonooohmmpTn^ 

~ 'O oooo o 



CS1 NO ^"OO N NO -+0O N NO "tOO <N NO ^-CO OJNO 'tOfONH iflOM 

On W ro -*no NOO h n "i-NO r-00 n ro 4- io t^.00 0* <-< w "4- vo r^oo On - 
m w i-i m -i m m cj N N N N « N <ororororororO'rt-^-Tj-T«-T*-Tj--*'£-io 
nonOnOnononOnOnonononononononononOnononOnOnO'OnOnonOnondnonCinO 



N 000NO ■<*• N 000 r^iOCOw O r^ vo CO m OMOMh OMfldH ONt^voro 
m ro 'S-no oo 6 N ro vo r 



N -*NO 00 N ""t-NO OC 



N cni ro ro ro r 



328 



FUNCTIONS OF A ONE-DEGREE CURVE. 



z 



Q 
< 



»OVO SO*0 m fO"^-iO^O\0 N -^"VO 00 N -^-VO 00 CO lO t^ N »O00 CO 
0*0 N oo ir>M NMOMflHOO "<i- O vo n Oimn n roO^O moo ioh t^ ro O vo 
m iovo vo t>.oo oo o>O\0 h h n ro co ■<»- ■<*• iovo vo tvoo ooononOhhncoco 

o N 0>o^o*o^ON^o>ONO>o>o\o>ffio\o>o\OM^oio>ON^osai^o\o>o*oiO' 



vo 00 N "*-vo On h rl-vo 0>« iflNO fOVO Ov N »O00 N iflO>« iflOTCNO ■"*■ 

VO m t^. N ^ N t^ COOO CO00 "^-OvTt-0 »o O 10 m vo w f*» n t^. rooo co o\ ■<*- 0° »o 

ro -<*■ ■«*• »o iovo vo r-> h*oo oo On Ov h h n cn ro ro rt- rf io iovo vo t*» t^oo ov on 

. n N N N N n n cs n N N N n rofOforOfonronrorofnnmforomfOfO 



P 
K 
O 



2 



o 
aa 
U 



55 
2 



m lO On -^-00 NVO mOvrOt^NVO ■* On CO »^ h VO o -*-oo mshvo ^-00 



CO N^N Ovt^"*N OvVO "«*■ W CO VO COO Ni«N On t>» •« 



■too U1M0 t>. "* H 



H ro -<J-VO 00 ON H CO iTjVO 00 N (OlflNOiO N ^-lONOH « "»fVO 00 On h CO 

t^ t^ t^ r^ t>. t^oo ooooooooononononononOOOOOOhi-ihmhhncn 
t^t**t-.t-.t^t^r^r-^t^t^t>.t^t^t^t^r>» t^oo oooooooooooooooooooooooooo 
OnOnOnOvOnOnOnOnOnOnOnOnOnOnOnOnOiOn.On On»On OnOnOnOnOnOnOnOnOnOv 



Q 



z 



O N -+VO CO o n ^t-vo 00 N -*-VO 00 « ^vo 00 N ^vo 00 N -4-VO 00 o 

MHMMHNNNNMfOtOmfOCOt'j-'J-'t'tlOiOiOiO IOVO 



Ov O^OO 00 t^ tvVO VO VO vo VO m IOVO VOVOVOVO N t-NOO CO O»O\0 H H N (Otm 
O\\r,H00 roOMOH ncoomoh ncoomoh NWOIOH !■>.■<$• bvo woo 4 6 
vo t^oo co oo\0 h h n n n*tio iovo o t^oo ooONOOt-iNNcoro'*i-m 

MWHHMMNNNNNNNNNNNCNNNNCNlCOrOCOCOCOrOCOCOrO 
OnOnOvOnOnOnOnOnOnOnOnOnOnOnOnOnONjOnOvOnOnOnOnOn.OnOnOnOnOno^On 



OOMHNNfO^-Tf- lOVO t>»00 OO H «+>0 t^cO h ro ^vo COON "*vo 

rooo' rooo' rooo rooo rooo rooo rooo -<1-On^-Ont|-ontJ-o "^ 6 io 6 io m vo h vo' 
oooo oonO m m n n roro-^-'^-io iovo vo t^ t^oo ooo O h h n cm com 

OOOOWWHHHMHMHMHHHHHMWMMNNNNNNNN 



vo "*00 N vo h io On co r». h vo ^-oo N vo m io on co t^ n vo O "*-oo cosh 

ro iovo' ts(>6 N n 4-vo NOO « h tJ-vo t^ on m ro tj-\o NO>0 h m tJ-no 
ONOvONOvOvOOOOOOOHt-tMMHMHCviCM<N(NCMNNrororororo 
vo <o vo vo vo t^t^t^t^t^t^t^t^t^t^t^t^f^t^t-^f^t-^t^f^t^t^^t^t^t^t^ 

«NMM«««NN«NNNNNC1CIN«NNNCI««NN««NCI 



vo ■**• N On f* ■**■ n smmOoo mroooo loroooovo roooo io co o oo mm 

00 6 N* rO IO f>. On H CM t}-VO 00 O « CO IOVO OOONCOiOt^ONOW -<J-VO NOvh 

m w N N N N cs rororororOfO"*-Ti-'<*-Tf-<i-ioioioiom iovo vo vo vo vo vo r>» 
OnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOn 



N *VO COON "«-VO 00 « "^-VO COON -«-vO CO O N "^-VO COON M-VO CO Q 

h m ii m h cn n n n cs mcommco<t*'t'>t't>oin»oio IOVO 



FUNCTIONS OF A ONE-DE&EEE CURVE. 



329 



vOHVOwVOHVOHVOHVONt*. COCO "<*• VO H SmOMflNOO ^* SMO t>» 

r>."4-0 smovo co o>vo n amN» ionoo io h oo *hoo ^hoo ij-moo -<*- 

N fl-ft mvO VO t>» t^cO On O m m N CO CO "<*- IO u-)VO tv t^OO On On m m N 
NNNNNNSNtststs t^OO COOOOOCOOOOOOOOOOOOOOOOOOOOO O On O On 

O v O^O^O^O^O | C | 0^0*CiO^O^O^C'O^O^O^O v OnO^OnO^CiO^O^O^O^O^&iO v O^ 



00 •<*- VO N 00 -*- vO mCMflNM iflNOO ION OnVO MhOO Ifl N MO N 

OnitjhvO eg NfOO\ t iO« NM00 *OMOm\0 N CO * CMO h In n 00 "t"6 
lOVO Cn t^OO 00 On O ih m N N CO ro -<1- tJ- iovo vO t>. t^OO OOOnOOmwcnICO 
u-> iO iO iO lO lO lO <0"0 VOVOVOVOVOVOVOVOVOVOVOVOVOVOVOVO t^ r>, r>» f^ In tv. 
lOiOiOlOtoiOU^iniOlOlOiOlOlOlOlOiOlOlOlOlOlOlOlOlOlOiOlOiniOiO 



oo wnh^o o <tOvfONNvo i 



iO On -*00 MNmvO N IO On COOO N t- h vo 



m CO »4-vO NOO h fO M-VO SO>0 h m tJ-vO N^O N « tJ-<0 n t> o N CO IO 
N N m n N N mmrofOcororoTt-'t'+M-^-^t-inioinioiOio iovo vo vo vo 

OOOOOOOOCOOOCOCOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 
NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN 



CO NMO t>» ■> 



■100 IO N 00 ION OniON OnvO CO On NO MO^O MONO COO f^ 



-tf-vo r- On 



N M-no SOv 



CO -^"O 00 On f 



. . N -^"0 NOnh N tJ-nO N(M- 
. NN r-^00 0000000000 OnOnOnOnOnOnO h 
OOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOO OnOnOnOnOnOnOnOnOnOnOOnOnOnOn 
OnOnOnOnOnOnOnOnOnOnOiOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOOnOnOn 



rONO On N IO00 H lOOO " 1O00 N IO On COVO "*00 N VO m to On CO00 N SN\Q 

VO N00 IO" t^-^-OVO COONlONOO ■«*- hi t>» tJ- VO CO OnVO MOO IO Hi 00 ■<*- H f>, 
CO -^ ■<*■ IOVO VO t^OO 00 OnOnO w w CM CO CO tJ- IO lOVO VO tNCO OO On O hi N N 
lOiOiOiOiOiOiOiOiOiO lOVO VOVOVOVOVOVOVOVOVOVOVOVOVOVO NSNNN 

ONONONONONONONONONOiONONONONONONONONONONONONO>ONONONO>ONONONON 



•^-00 N VO -^-00 CO tv. h vo IO0 '^■ON'«» , ON , ^-ON'<*-ONThONiO0vO w SN00 

IO VO m NN tN CO00 -^-OnioOvo mvO N tv rO00 ^ On iO VO N tN rOOO 4 w 
ooOhmwncoco-^ti- iovo vo tN 1^.00 oo ono>0 O m n n ro n ■+ -t io id 
ro-<i--^--^-Ti--<t--<!j-Tj-Th , >^Ti--<i--«*--<!i-^-^-Tj-Tj--^-Thioio»oioioioioioioio>o 

1010»0101010 10»010»0»0«01010»010»010»010»010>010>01010»OIOXO»0 



00 CO t>» m vo -*00 fONHVO '^•ONCOt^NVO «0 0\ COOO N VO t-i io On ^oo 

oo 6 h co -^vo t^co m co M-vo t^oo m co tj-vo t^oo m co -<*-vo t^oo' o' M 
fN,co ooooMooteMo>o>o\o>(>CMJOOOOoooHHHWHHMciti 
tv.tN.tN tN,rNtN.t>.tN.fN.fN,tN,tN,fN,tN» t^. oo ooooooooooooooooooooooooooooco 

NNNNNNNNNNNNNNNNNNNNNNNNNNNCMNNN 



I OnVO COO N + H00VO COO S ■+ H 00 »ON OnVO COO S ■* H 00 »ON f>VO CO 





u 



CO -<f VO 00 hi CO lOVO 00 N CO IO t^OO N fO IO N On O N -t IO N Q> N ■*■ 

N N N N roroforoMfO-t't^^'+'tioioiomuN iovo vovo vovovo nnk 

OOCOCOCOOOOOaJTOOOOOTOCOCXDOOOOOOOOOOOOCOOOCOOOOOOOOOOOOOOOOOCO 
OnOOnOnO>OnOnOnOnOnOnOnOnOnOnOnOiOnOnOnOnO>OnOnOnOnOnOnOnOnOn 



N ^"VO CO N iJ-vo 00 N "^-VO CO N "ii-vO 00 N ^-VO 00 N -<*-\0 CO O 

H h m m m n n n n n cocococorO'«*'<-*r'«"<*'iom«nu-> \n\& 



TABLE XVII. 
RISE PER MILE OF VARIOUS GRADES. 



332 TABLE XVII.-RISE PER MILE OF VARIOUS GRADES. 



Rise 

per 

Cent. 


Feet per 
Mile. 


Rise 
per 
Cent. 


Feet per 
Mile. 


Rise 
Cent. 


Feet pet 
Mile. 


Rise 

per 

Cent. 


Feet per 
Mile. 


•ox 


.528 


.61 


32.208 


1. 21 


63.888 


1.81 


95-568 


•02 


1.056 


.62 


32.736 


1.22 


64.416 


1.82 


96.096 


.03 


1.584 


.63 


33-264 


1.23 


64.944 


1.83 


96.624 


.04 


2.X12 


.64 


33.792 


1.24 


65.472 


1.84 


97.152 


.05 


2.640 


.65 


34.320 


1.25 


66.000 


1.85 


97.680 


.06 


3.168 


.66 


34.848 


1.26 


66.528 


1.86 


98.208 


•°z 


3.696 


.67 


35.376 


1.27 


67.056 


1.87 


98.736 


.08 


4.224 


.68 


35.904 


1.28 


67.584 


1.88 


99.264 


.09 


4.752 


.69 


36.432 


1.29 


68.112 


1.89 


99.792 


•xo 


5.280 


.70 


36.960 


1.30 


68.640 


1.90 


100.320 


• XX 


5.808 


•7i 


37.488 


1 -3i 


69.168 


1. 91 


100.848 


• 12 


6.336 


.72 


38.016 


1.32 


69.696 


1.92 


101.376 


•13 


6.864 


•73 


38.544 


i«33 


70.224 


i-93 


101.904 


•14 


7-392 


•74 


39.072 


i-34 


70.752 


1.94 


102.432 


•15 


8.448 


•75 


39.600 


1-35 


71.280 


1-95 


102.960 


.16 


.76 


40.128 


1.36 


71.808 


1.96 


103.488 


:3 


8.976 


•77 


40.656 


1.37 


72.336 


1.97 


104.016 


9.504 


.78 


41.184 


1.38 


72.864 


1.98 


104.544 


.19 


10.032 


•79 


41.712 


1-39 


73.392 


1.99 


105.072 


•20 


10.560 


.80 


42.240 


1.40 


73.920 


2.00 


105.600 


•21 


11.088 


.81 


42.768 


1.41 


74.448 


2.10 


110.880 


.22 


XI. 616 


.82 


43.296 


1.42 


74.976 


2.20 


116. 160 


.23 


X2.144 


.83 


43.824 


1.43 


75.504 


2.30 


121.440 


.24 


12.672 


.84 


44-352 


1.44 


76.032 


2.40 


126.720 


.25 


13.200 


.85 


44.880 


1.45 - 


76.560 


2.50 


132.000 


.26 


13.728 


.86 


45.408 


1.46 


77.088 


2.60 


137.280 


.27 


14.256 


.87 


45.936 


1.47 


77.616 


2.70 


142.560 


.28 


14.784 


.88 


46.464 


1.48 


78.144 


2.80 


147.840 


.29 


15.312 


.89 


46.992 


1.49 


78.672 


2.90 


153.120 


.30 


15.840 


.90 


47.520 


1.50 


79.200 


3-00 


158.400 


•31 


16.368 


.91 


48.048 


i-5i 


79.728 


3.10 


163.680 


.32 


16.896 


.92 


48.576 


1.52 


80.256 


3.20 


168.960 


•33 


17.424 


•93 


49.104 


i-53 


80.784 


3-3o 


174.240 


•34 


17.952 


•94 


49-632 


1-54 


81.312 


3.40 


179.520 


•35 


18.480 


•95 


- 50.160 


i-55 


81.840 


3.50 


184.800 


.36 


19.008 


.96 


50.688 


1.56 


82.368 


3.60 


190.080 


•37 


19.536 


•97 


51.216 


1-57 


82.896 


3.70 


195.360 


.38 


20.064 


.98 


51.744 


1.58 


83.424 


3.80 


200.640 


•39 


20.592 


•99 


52.272 


1.59 


83-952 


3-90 


205.920 


•40 


21.120 


1. 00 


52.800 


1.60 


84.480 


4.00 


211.200 


•4i 


21.648 


1. 01 


53.328 


1. 61 


85.008 


4.10 


216.480 


.42 


22.176 


x.02 


53.856 


1.62 


85.536 


4.20 


221.760 


•43 


22.704 


1.03 


54.384 


1.63 


86.064 


4-30 


227.040 


•44 


23.232 


1.04 


54-9 I 2 


1.64 


86.592 


4.40 


232.320 


•45 


23.760 


1.05 


55.440 


1.65 


87.120 


4-50 


237.600 


.46 


24.288 


1.06 


55.968 


1.66 


87.648 


4.60 


242.880 


•47 


24.816 


1.07 


56.496 


1.67 


88.176 


4-70 


248.160' 


.48 


25.344 


1.08 


57.024 


1.68 


88.704 


4.80 


253-440 


•49 


25.872 


1.09 


57.552 


1.69 


89.232 


4.90 


258.720 


.50 


26.400 


X.IO 


58.080 


1.70 


89 . 760 


5.00 


264.000 


•Si 


26.928 


1. 11 


58.608 


1.71 


90.288 


5.10 


269.280 


•52 


27.456 


1. 12 


59.136 


1.72 


90.816 


5.20 


274.560 


•53 


27.984 


i-*3 


59.664 


1-73 


91-344 


5.30 


279.840 


•54 


28.512 


1. 14 


60. 192 


1.74 


91.872 


5-40 


285.120 


•56 


29.040 


x-15 


60.720 


1.75 


92.400 


5-5C 


290.400 


29.568 


1. 16 


61.248 


1.76 


92.928 


5.60 


295.680 


•57 


30.096 


1. 17 


61.776 


1.77 


93.456 


5.70 . 


300.960 


.58 


30.624 


x.x8 


62.304 


1.78 


93.984 


5.80 


306.240 


:g 


31.152 


x.19 


62.832 


;:e 


94.512 


5-9° 


311.520 


31.680 


1.20 


63.360 


95.040 


6.00 


316.800 



ADDENDA 






ADDENDA, 



TABLE 

OP 
FEET, INCHES AN ) RECIPROCALS OP VARIOUS TRACK GAUGES. 



6' 


72" 


.013389 


Metre 


39.375" 


.0254 


5' 


60" 


.01667 


3' 6" 


42" 


.0233 


4' 9" 


57" 


.01754 


3' 4" 


40" 


.0250 


4' 8f ' 


56f 


.01769 


30 


36" 


.02^8 



TABLE 



MINUTES OP A DEGREE EXPRESSED IN DECIMALS. 



1 


.0167 


16 


.2667 


31 


.5167 


46 


.7667 


2 


.0333 


17 


.2833 


32 


.5333 


47 


.7833 


3 


.0500 


18 


.3000 


33 


.5500 


48 


.8000 


4 


.0667 


19 


.3167 


34 


.5667 


49 


.8167 


5 


.0833 


20 


3333 


35 


.5833 


50 


8333 


6 


.1000 


21 


.3500 


36 


.6000 


51 


.8500 


7 


.1167 


22 


.3667 


37 


.6167 


52 


.8667 


8 


.1333 


23 


.3833 


38 


.6333 


53 


.8833 


9 


.1500 


24 


.4000 


39 


.6500 


54 


.9000 


10 


.1667 


25 


.4167 


40 


.6667 


55 


.9167 


11 


.1833 


26 


.4333 


41 


.6833 


56 


.9333 


12 


.2000 


27 


.4500 


42 


.7000 


57 


.9540 


13 


.2167 


28 


.4667 


43 


.7167 


58 


.9667 


14 


.2333 


29 


.4833 


44 


.7333 


59 


.9833 


15 


.2500 


30 


.5000 


45 


.7500 


60 


1.0000 



336 



ADDENDA. 



CONDENSED TABLE OF RADII 

INCLUDING SHORT CHORDS. 



Degree 








op 


100' Chord. 


50' Chord. 


25' Chord. 


Curve. 








1° 


5729.66 


5729.60 




2° 


2864.93 


2864.82 




3° 


1910.08 


1909.91 




4° 


1432.69 


1432.47 




5° 


1146.28 


1146.01 




6° 


955.37 


955.04 




7° 


819.02 


818.64 




8° 


716.78 


716.34 




9° 


637.28 


636.78 




10° 


573.69 


573.14 




11° 


521.67 


521.07 




12° 


478.34 


477.68 




13° 


441.68 


440.97 




14° 


410.28 


409.51 




15° 


383.06 


382.25 




16° 


359.26 




358.17 


17° 


338.27 




337.11 


18° 


319.62 




318.46 


19° 


302.94 




30164 


20° 


2S7.94 




286.57 


21° 


274.37 




272.93 


22° 


262.04 




260.54 


23° 


250.79 




249.22 


24° 


240.49 




238.84 


25° 


235.65 




229.30 


• 26° 


222.27 




220.49 


27° 


214.18 


p 


212.30 


28° 


206.68 




204.76 


29° 


199.70 




197.70 


30° 


193.60 




190.79 



ADDENDA. 



337 



To express gradients per cent, (page 332), in angular meas- 
ure, multiply the rate per cent, by 34.3; the product will be 
given in minutes of a degree. 



EXAMPLE. 



Gradient 

PER CENT. 


Minutes. 


Gkadient 

PER CENT. 


Minutes. 


.20 


6.86 


2.50 


85.75 


.40 


13.72 


3.00 


102.90 


.60 


20.58 


3.50 


120.005 


.80 


27.44 


4. 


137.20 


1. 


4.30 


4.50 


154.35 


1.20 


41.16 


5. 


171.5 


1.40 


48.02 


6. 


205.8 


1.60 


54.88 






1.80 


61.74 






2. 


68 60 







If the gradient per cent, be multiplied by 57.14, the result 
will be expressed in hundredths of a degree. 

SOUND. 

At freezing temperature, 32 degrees Fahrenheit, in calm air, 
the velocity of sound may be assumed 1100 feet per second. 
For lower temperatures subtract, and for higher add, a half 
foot per degree. 

The intensity of sound varies inversely as the square of the . 
distance. The velocity varies directly as the temperature. 
It is nearly four times as great in water as in air; and in wood 
ten to sixteen times as great. 



AMERICAN AND FRENCH EQUIVALENTS. 

LINEAR MEASURE. 

1 inch = 2.54 centimetres ; 1 centimetre = .394 inches. 
1 foot = .3048 metres: 1 metre = 3.2809 feet. 
1 yard = 3 feet = .9144 metres; 1 metre = 1.0936 yards. 
' 1 rod = 16.5 feet = 5.029 metres; 1 metre = 0.2 rods. 



338 ADDENDA. 

1 surveyor's chain = 66 feet = 4 rods = 20.117 metres; 

1 metre = .05 chains. 
1 kilometre = .6214 miles = 3281 feet. 
1 statute mile = 5280 feet = 80 rods = 1.6093 kilometres 



AMERICAN AND FRENCH EQUIVALENTS 

SQUARE MEASURE. 

1 square inch = 6.4515 square centimetres. 

1 square centimetre = 0.1550 square inches. 

1 square foot = 0.929 square metres. 

1 square metre — 1.19659 square yards. 

1 square acre = 43560 square feet = 4840 square yards. 

1 square hectare = 2.4711 acres = 11960 square yards. 

1 acre = 0.4047 hectares. 

1 square kilometre = .3861 square miles. 

1 square mile = 2.5899 square kilometres. 

1 square rod = 272.25 square feet = .00259 hectares. 

AMERICAN AND FRENCH EQUIVALENTS. 

CUBIC MEASURE. 

1 cubic inch = 16.383 cubic centimetres. 
1 cubic centimetre = .0610 cubic inches. 
1 cubic foot = 28.316 cubic decimetres. 
1 cubic decimetre = .0353 cubic feet. 
1 cubic yard == .7645 cubic metres. 
1 cubic metre = 1.308 cubic yards. 



INDEX. 



PAGE 

Abbreviations explained • ix 

Acres, roods, and perches in square feet, Table VI 152 

Adjustment and use of instruments 23 

Angles of frogs, to find 129 

index, to find 69 

intersection, to find 55 

plane 12 

to read on verniers 43 

tangential and deflection . 50 

of switch-rails 130 

Apex distance of curves, to find 52 

Arc, functions of, to find . 13 

Arithmetical complement 6 

Axemen, duties of 84 

Azimuths of North Star, Table IL 150 

Barometer, levelling by -, 29 

Bench-marks, proper intervals for 83 

Bubble, to adjust on level 25 

to adjust on transit 40 

Chain, to lay out curves with 63 

Chainman, duties of 42 

Chief engineer, duties of 79 

Chords, to calculate 54, 58 

Table XVI 269 

Circle, propositions concerning 49 

Circular arcs to radius of 1, Table VI. 152 

Complement of an angle • 12 

arithmetical • 6 

Compound curves. See Curves. 

Contour maps, utility of 85 

Correction for curvature and refraction in levelling 28 

Cosines defined 12 

Crossings, plain rules for laying off 139 

Cross-hairs, to adjust 24, 26, 40 

eccentricity of 24 

to put in new 44 

339 



340 INDEX, 

PAGE 

Cross-sectioning. See Slope stakes. 

Cubes and cube roots of numbers, Table XI 161 

Curves, circular, on railroad defined . . 51 

to find radius, length, degree, apex distance, chord, mid- 

ordinate, and external secant 52, 56 

form for field notes 70 

Curves, how to lay out on the ground, — 

with the chain only 63 

with transit and chain 66 

hints as to field-work 8i 

protractor for 84 

slackening grade on 87 

terminal 88 

Curves, simple, location of, — 

how to proceed when the P. C. is inaccessible 93 

to shift the P. C. in order to strike a fixed tangent 96 

to change radius from same P. C. in order to strike a fixed tan- 
gent 97 

to triangulate on 94 

to pass through a fixed point 127,128 

Curves, compound, — 

how to proceed when the P. C. C. is inaccessible 95 

to compound a curve in order to strike a fixed tangent .... 98 

to shift a P. C. C. in order to strike a fixed tangent . . . . . 99 

summary of rules for 101 

to compound into a tangent intersecting main curve on concave 

side 102 

to compound into a tangent intersecting main curve on convex 

side 10 

Curves, reversed, — 

parallel tangents, radii equal 115 

parallel tangents, radii unequal 117 

angles unequal, radii equal 119 

angles unequal, tangent points fixed, radii equal 120 

divergent tangents, radii equal, advancing towards intersection . 123 

receding from intersection 124 

to shift a P. R. C. in order to strike a fixed tangent 125 

Curves, miscellaneous, — 

elevation of outer rail 141, 142 

degree of, to find by calculation 52, 55 

to find on ground 145, 146 

to connect curves of contrary flexure by short tangents ... 89 

to locate a Y from a tangent 103 

from a convex curve 104 

from a concave curve 106 

to locate a tangent to a curve from a fixed point 108 

to two curves already located 109 

to substitute a curve for a tangent connecting two curves ... 109 

terminal curves * 88 



INDEX. 341 

PAGE 

Curves, miscellaneous — con tinned. 

trackmen's table of curves and spring of rails 143 

vertical curves, to calculate 36 

to project 39 

Datum in levelling 27 

Decimals of an acre per 100 feet for various widths, Table IV. . . . 151 

Deflection angles and distances explained 50 

to rind 57, 64, 68 

short rule for sub-deflections 68 

limit in field-practice 82 

Degree of curve, to calculate 52, 55 

to find on ground 145, 146 

Deviations from project admissible on location 81 

Distances, tangential and deflection, defined 50 

table of 155 

of frogs from toe of switch 130, 132 

tables of 135, 136 

Elevation of outer rail on curves 141 

table of 142 

Excavation and embankment, to stake out 30 

External secants, to find 54 

of a 1° curve, Table XVI 269 

Extreme elongations of North Star, Table 1 148 

Feet in decimals of a mile, Table VII 153 

I Field-work, suggestions concerning 79, 85 

Field-book, form of, for level . . . 27 

for transit 70 

for slope stakes 33, 34, 35 

Frogs and switches 129 

rules for angles and distances 130 

table of, switch-rails straight 135 

switch-rails curved 136 

plain rules for locating, switch-rails straight 132 

switch-rails curved 133 

on narrow gauges 134 

patterns for 134 

Functions, trigonometrical, defined 12 

logarithmic, of arcs, to find 14 

eneral propositions in trigonometry „ 15 

as to circles 49 

Jrade, to slacken on curves » 87 

rise per mile, Table XVII 332 

>rade lines, how to project on map . 86 

how to trc ce in field 81 

Heights, to find by barometer and thermometer . . „ «, e o « o 29 



342 INDEX. 

pag: 

Inches in decimals of a foot, Table VIK 153 

Index angles, to determine 

Instruments, adjustment and use of 23 

Intersection angles of tangents, to find 55 

desirable to fix on ground 

Level, to adjust 24 

Leveller, duties of 

Levelling, art of 26 

by barometer and thermometer 29 

correction for curvature and refraction ' . . . 28 

form for field-book 27 

rules for exact work 27 

rules for survey and location 28 

suggestions concerning 83 

Location, problems in field 94 

admissible errors on ground 81 

form of record for 81 

projects, hints concerning 84 

of terminal curves 88 

of a Y ....... 103, 104, 106 

Logarithms explained 3 

multiplication by 5 

division by 6 

of numbers, to find 4 

Table XII 179 

roots and powers by 7 

Logarithmic sines, tangents, &c, to find 13 

table of, XIII W 

Maps, contour, utility of 85 

notes for 82, 83 

not sufficient for intelligent projects 79 

Meridian, to establish 44 

by equal shadows 45 

by North Star 45 

times of passage of North Star, Table 1 148 

Multiplication by logarithms 5 

Natural sines, tangents, &c, defined 12 

Table XIV. 243,256 

Needle, magnetic, to adjust . 41 

to re-magnetize 44 

hints as to management 44 

bearings should always be noted 82 

North Star, to establish meridian by • . . 45 

times of meridian passage, Table 1 148 

extreme elongations of, Table 1 148 

azimuths and natural tangents, Table II. ... - 150 



IXDEX. 



343 



PAGE 

Obstacles in the field to vision 73 

to measurement 74 

Ordinates of circular curves, to find 58, 59 

of parabola, to find 36 

of a 1° curve, Table X 155 

Parabola, ordinates of 36, 59 

Plane trigonometry 12 

Powers and roots of numbers by logarithms 7 

Propositions, general, in, trigonometry 15 

Protractor for curves described 84 

how to make * 85 

Rails, table of spring for trackmen 143 

Radius of a curve, how to find 52, 54, 56 

of a turnout curve 129 

plain rule for, on curves „ 133 

for narrow gauges 134 

Radii and their logarithms, Table IX 155 

Records, forms for 81 

Refraction and curvature, correction for 28 

Reversed curves. See Curves. 

Rise per mile of various grades, Table XVII 332 

Rod, levelling 28 

how to read 42 

Rodman, duties of 83 

Roods and perches in decimals of an acre, Table III 151 

Roots and powers of numbers by logarithms ......... 7 

Senior assistant, duties of 80 

equipment for 81 

Sines defined 12 

Shadows, to fix true north by 44 

Slopes for topography, Table XV 268 

Slopeman, duties of 84 

Slope stakes, to set 30 

for earth excavation 31 

for embankment • 33 

for hillsides and rock 35 

field record of work „ 34 

Spring of rails, table for trackmen 143 

Squares, cubes, and roots of numbers, Table XI 161 

Supplement of an angle 12 

Survey, form for record 81 

to facilitate . * '. . . . . 82 

Switch-rails, angles of 130 

tables of . . . . 135, 136 

Tangent, or apex distance of curve, to find , 62, 54 



344 



INDEX. 



PAGE 

Tangent of a 1° curve, Table XVI 269 

to curve from a fixed point, how to locate 108 

to two curves on the ground, how to locate ........ 109 

Tangential angles and distances explained . - . 50 

how to find 57, 58, G4 

Thermometer, levelling by 29 

Track problems 115 

Trackmen's plain rules for finding frog distances 132,133 

tables of turnouts 135, 136 

plain rules for laying off turnouts with tape-measure and pins . 137 

crossings on straight lines and on curves 139 

elevation of outer rail 142 

instructions how to put in missing stakes on curves with tape- 
measure 144 

table of curves and spring of rails 143 

explanation of the trackmen's tables 144 

how to find the degree of a curve 145, 146 

Transit, adjustment of 40 

cross-hairs 24 

Transitman, duties of 82 

Triangles, solution of, — 

two angles and a side given 16 

two sides and an angle given 17 

three sides given 18 

Triangles, right-angled, solution of 19 

Trigonometry, plane 12 

general propositions 15 

Turnouts. See Trackmen. 

Vernier explained 42 

on transit 43 

Versed sines defined « . 12 

to calculate 54, 58 

of a 1° curve, Table XVI 269 

Vertical curves, to calculate „ 36 

to project » • • • 39 



Tables : 

Ordinates of a 1° curve ....... 60 

For locating terminal curves . 88 

Tangents between curves of contrary flexure 89 

Turnouts, switch-rails straight 135 

switch-rails curved 136 

Elevation of outer rail on curves 142 

Curves and spring of rails 143 

I. Culminations and elongations of North Star 148 



INDEX. 345 

PAGE 

II. Azimuths of North Star, and their natural tangents .... 150 

III. Roods and perches in decimal parts of an acre 151 

IV. Decimals of an acre in one chain length of 100 feet, and of 

various widths 151 

V. Acres, roods, and perches in square feet 152 

VI. Circular arcs to radius of 1 152 

VII. Feet in decimals of a mile 153 

VIII. Inches reduced to decimal parts of a foot 153 

IX. Radii and their logarithms, middle ordinates, and deflection 

distances 155 

X. Metric-curve table 159 

XL Squares, cubes, roots, and reciprocals of numbers, from 1 

to 1,042 161 

XIL Logarithms of numbers from 1 to 10,000 179 

XIII. Logarithmic sines, cosines, tangents, and cotangents .... 197 

XIV. Natural sines and cosines 243 

Natural tangents and cotangents 256 

XV. Slopes for topography 208 

XVI. Functions of a 1° curve 269 

XVII. Rise per mile of various grades 332 

ADDENDA. 

Table of feet, inches, and reciprocals of various track gauges .... 335 

Table of minutes of a degree expressed in decimals 335 

Condensed table of radii including short chords 336 

To express gradients per cent, in angular measure ........ 337 

Sound 337 

American and French equivalents — Linear measure 337 

American and French equivalents— Square measure 338 

American and French equivalents— Cubic measure 338 



. 



